Computations using a polychronous wave propagation system

ABSTRACT

The present invention relates to a polychronous wave propagation system that is based on relative timing between two or more propagated waves through a wave propagation medium. The relative timing may be associated with interference patterns of energy between the propagated waves. Operational behavior of the polychronous wave propagation system is based on the relative timing of the propagated waves and distances between initiators that transmit the propagated waves and responders that receive the propagated waves. The operational behavior may include arithmetical computations, memory storage, Boolean functions, frequency-based computations, or the like. The polychronous wave propagation system relies on time delays between the propagated waves that result from propagation velocities of the propagated waves through the wave propagation medium. By incorporating the time delays into the system, operational capacity may be greatly enhanced.

RELATED APPLICATIONS

The present application is a continuation-in-part of U.S. patentapplication Ser. No. 12/826,146, filed Jun. 29, 2010, entitled“POLYCHRONOUS WAVE PROPAGATION SYSTEM,” which is a continuation-in-partof U.S. patent application Ser. No. 12/484,615, filed Jun. 15, 2009,entitled “POLYCHRONOUS WAVE PROPAGATION SYSTEM,” which claims thebenefit of U.S. Provisional Patent Application Ser. No. 61/061,436,filed Jun. 13, 2008, all of which are hereby incorporated herein byreference in their entireties.

This invention was made with government support under HR0011-07-3-0002awarded by DARPA. The government may have certain rights in thisinvention.

FIELD OF THE INVENTION

Embodiments of the present invention relate to autonomous asynchronouscomputing systems, which relate timing of asynchronous activity signalsto one another to perform useful functions, such as memory storage,Boolean functions, frequency-based computations, or the like.

BACKGROUND OF THE INVENTION

In researching theories of the brain, certain operational fundamentals,such as memory registration and recall, associative memory, and patternrecognition, suggest much greater complexity than what the basicunderlying physical structure provides. For example, a human brain canstore more information than would be expected given the number ofsynapses in the human brain. Therefore, human brain memory is more thanjust synaptic memory. Other factors come into play in conjunction withthe synapses to store all of the information. Since the brain operatesin an autonomous asynchronous environment, one theory is that timedelays may be important.

For example, if two neurons fire at a common post synaptic target, theirspikes travel along axons to the target, and if their spikes arrivesimultaneously at the target, a stronger response may be evoked than iftheir spikes arrive separately. However, axons have propagationvelocities that introduce conduction delays; therefore, both thedistances from the neurons to the target and the firing times of theneuron spikes determine when the spikes arrive at their target. Thepropagation velocity may be about one millimeter per millisecond formyelinated fibers and about one-hundred micrometers per millisecond fornon-myelinated fibers. Specifically, if a conduction path from a firstneuron to the target is about ten millimeters long and a conduction pathfrom a second neuron to the target is about two millimeters long, thefirst neuron will have to fire about eight milliseconds before thesecond neuron fires in order for both spikes to arrive simultaneously.

In general, neurons in spiking networks with conduction delays may firewith certain time-locked asynchronous patterns so that their spikes mayarrive at targets simultaneously. The additional dimension of timedelays in a brain may significantly increase a brain's capacity torepresent and process information. Such an activity may be calledpolychrony. Polychrony may be derived from poly, meaning many, andchronos, meaning time or clock. With an appropriate type of spike-timingdependent synaptic plasticity, spiking networks may self-organize andgenerate such polychronous activity, which may have relevance to memory,binding and gamma rhythms, mechanisms of attention, pattern recognition,and the like. Polychronous activity in the brain depends on specificityof synaptic connections, geometry and dimensions of axonal fibers,activity-dependent propagation velocities, dynamics of variousneurotransmitters, spike-generation mechanisms of neurons, and otherbiological factors.

Applying polychronous techniques to physical systems, electronicsystems, or both may significantly increase the capacities, thefunctionalities, or both of such systems. Such systems may operate atmuch higher frequencies than a brain, which may operate in a frequencyrange up to about 100 hertz. For example, networks having time delayscan encompass greater functionalities than comparable networks withouttime delays. A dynamic system having a given number of state variablesmay be represented by a differential equation that has a solution spaceof the same dimension as the number of state variables. However, whenasynchronous time delays are added to the dynamic system, adifferential-delay equation that is representative of the dynamic systemwith delays has an infinite dimensional solution space. Thus, there maybe significant benefits from a polychronous physical or electronicsystem, such as arithmetical computations.

SUMMARY OF THE EMBODIMENTS

The present invention relates to a polychronous wave propagation systemthat is based on relative timing between two or more propagated wavesthrough a wave propagation medium. The relative timing may be associatedwith interference patterns of energy between the propagated waves.Operational behavior of the polychronous wave propagation system isbased on the relative timing of the propagated waves and distancesbetween initiators that transmit the propagated waves and respondersthat receive the propagated waves. The operational behavior may includearithmetical computations, memory storage, Boolean functions,frequency-based computations, or the like. The polychronous wavepropagation system relies on time delays between the propagated wavesthat result from propagation velocities of the propagated waves throughthe wave propagation medium. By incorporating the time delays into thesystem, operational capacity may be greatly enhanced.

In one embodiment of the present invention, the propagated wave may be apulsed wave lasting just a few cycles. Further, a responder and aninitiator may be combined to form a transponder, which may receive twoor more propagated waves and initiate a response transmission of apropagated wave based on the relative timing between the receivedpropagated waves. In an exemplary embodiment of the present invention,the response transmission is initiated based on simultaneous receptionof the received propagated waves. In general, the relative timing may beassociated with interference patterns of energy between the receivedpropagated waves. The transponder may include a controllable oscillatorblock, which may be controlled based on the responder section of thetransponder and may provide a controlled signal upon which the responsetransmission is based. In one embodiment of the controllable oscillatorblock, the controllable oscillator block oscillates at a controllablefrequency, which may be based on frequencies of received pulsedpropagated waves. The initiated transmission may be a pulsed propagatedwave that may be at the frequency of the controllable oscillator block.

Those skilled in the art will appreciate the scope of the presentinvention and realize additional aspects thereof after reading thefollowing detailed description of the preferred embodiments inassociation with the accompanying drawing figures.

BRIEF DESCRIPTION OF THE DRAWING FIGURES

The accompanying drawing figures incorporated in and forming a part ofthis specification illustrate several aspects of the invention, andtogether with the description serve to explain the principles of theinvention.

FIGS. 1A through 1D show a first initiator transmitting a first wavethat propagates through a wave propagation medium.

FIGS. 2A through 2H show a polychronous wave propagation system, inwhich asynchronous waves that have timing relationships to one anotherare transmitted and propagate through the wave propagation medium.

FIG. 3 is a three-dimensional graph showing space-time relationships ofthe polychronous wave propagation system illustrated in FIGS. 2A through2H.

FIG. 4 shows details of the polychronous wave propagation systemillustrated in FIGS. 2A through 2H, according to one embodiment of thepolychronous wave propagation system.

FIG. 5 shows details of a first, a second, a third, and a fourthtransponder illustrated in FIG. 4, according to one embodiment of thepolychronous wave propagation system.

FIG. 6 shows details of the first transponder illustrated in FIG. 5,according to one embodiment of the first transponder.

FIG. 7 shows details of a first responder and a third initiatorillustrated in FIG. 6, according to one embodiment of the firsttransponder.

FIG. 8 shows details of first transmit circuitry illustrated in FIG. 7,according to one embodiment of the first transmit circuitry.

FIG. 9 shows details of the polychronous wave propagation system,according to an alternate embodiment of the polychronous wavepropagation system.

FIG. 10 shows details of a system array of initiators, responders, andtransponders.

FIGS. 11A through 11F and 12A through 12G illustrate differentapproaches to defining TRUE and FALSE states, which are used by thepolychronous wave propagation system to provide Boolean functionality.

FIGS. 13A through 13E illustrate behavior of a Boolean NOR gate usingthe polychronous wave propagation system with both A and B inputs of theNOR gate being in a TRUE state.

FIGS. 14A through 14F illustrate behavior of the Boolean NOR gate usingthe polychronous wave propagation system with both A and B inputs of theNOR gate being in a FALSE state.

FIGS. 15A through 15E illustrate behavior of the Boolean NOR gate usingthe polychronous wave propagation system with the A input of the NORgate being in a TRUE state and the B input of the NOR gate being in aFALSE state.

FIGS. 16A through 16E illustrate behavior of the Boolean NOR gate usingthe polychronous wave propagation system with the A input of the NORgate being in a FALSE state and the B input of the NOR gate being in aTRUE state.

FIGS. 17A through 17E illustrate behavior of a four transponderreverberating memory cell in a TRUE state using the polychronous wavepropagation system in one embodiment of the polychronous wavepropagation system.

FIGS. 18A through 18D illustrate behavior of the four transponderreverberating memory cell in a FALSE state using the polychronous wavepropagation system in one embodiment of the polychronous wavepropagation system.

FIG. 19 illustrates behavior of a reverberating memory doublet cellusing the polychronous wave propagation system in an alternateembodiment of the polychronous wave propagation system.

FIGS. 20A and 20B illustrate behavior of a three transponderreverberating memory cell using the polychronous wave propagation systemin an additional embodiment of the polychronous wave propagation system.

FIG. 21 illustrates frequency detection behavior of the polychronouswave propagation system in another embodiment of the polychronous wavepropagation system.

FIG. 22 shows a first controllable oscillator block, which forms part ofthe first transmit circuitry illustrated in FIG. 8.

FIG. 23 shows details of the first controllable oscillator blockillustrated in FIG. 22, according to one embodiment of the firstcontrollable oscillator block.

FIG. 24 shows details of a voltage controlled oscillator neuron (VCON)oscillator illustrated in FIG. 23.

FIG. 25 shows a graph illustrating first and second spatial variables ofa line segment between first and second transponders illustrated in FIG.21 versus time according to one embodiment of the polychronous wavepropagation system illustrated in FIG. 21.

FIG. 26 shows a graph illustrating third and fourth spatial variables ofthe line segment between the first and the second transpondersillustrated in FIG. 21 versus time according to an alternate embodimentof the polychronous wave propagation system illustrated in FIG. 21.

FIG. 27 shows a graph illustrating fifth and sixth spatial variables ofthe line segment between the first and the second transpondersillustrated in FIG. 21 versus time according to an additional embodimentof the polychronous wave propagation system illustrated in FIG. 21.

FIG. 28 shows a graph illustrating a combination of the first, thesecond, the third, the fourth, the fifth and the sixth spatial variablesillustrated in FIGS. 25-27 according to another embodiment of thepolychronous wave propagation system illustrated in FIG. 21.

FIG. 29 shows a graph illustrating a combination of the third, thefourth, the fifth and the sixth spatial variables illustrated in FIG. 28according to one embodiment of the polychronous wave propagation systemillustrated in FIG. 21.

FIG. 30 shows a graph illustrating a combination of the third, thefourth, the fifth and the sixth spatial variables illustrated in FIG. 28according to an alternate embodiment of the polychronous wavepropagation system illustrated in FIG. 21.

FIG. 31 shows a graph illustrating a combination of the third, thefourth, the fifth and the sixth spatial variables illustrated in FIG. 28according to an additional embodiment of the polychronous wavepropagation system illustrated in FIG. 21.

FIG. 32 shows a graph illustrating a combination of the first and thesecond spatial variables illustrated in FIGS. 25 and 26, andillustrating seventh and eighth spatial variables of the line segmentbetween the first and the second transponders illustrated in FIG. 21versus time according to an alternate embodiment of the polychronouswave propagation system illustrated in FIG. 21.

FIG. 33 shows a polychronous wave propagation system that is capable ofperforming and storing the results of arithmetic computations accordingto one embodiment of the polychronous wave propagation system.

FIG. 34 shows two sets of waves transmitted from transpondersillustrated in FIG. 33 according to an exemplary embodiment of thepolychronous wave propagation system.

FIG. 35 shows a set of waves transmitted from transponders intransponder arrays illustrated in FIG. 33 that identify results of asubtraction illustrated in FIG. 34.

FIG. 36 shows a three-dimensional view of the polychronous wavepropagation system having a group of wave propagation media according toone embodiment of the polychronous wave propagation system.

FIG. 37 shows details of a second wave propagation medium illustrated inFIG. 36 according to one embodiment of the second wave propagationmedium.

FIG. 38 shows details of a third wave propagation medium illustrated inFIG. 36 according to one embodiment of the third wave propagationmedium.

FIG. 39 shows details of an N^(TH) wave propagation medium illustratedin FIG. 36 according to one embodiment of the N^(TH) wave propagationmedium.

FIG. 40 shows a set of waves transmitted from transponders illustratedin FIG. 33 according to an exemplary embodiment of the polychronous wavepropagation system.

FIG. 41 shows a set of waves transmitted from transponders intransponder arrays illustrated in FIG. 40 that identify results of adivision by two.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The embodiments set forth below represent the necessary information toenable those skilled in the art to practice the invention and illustratethe best mode of practicing the invention. Upon reading the followingdescription in light of the accompanying drawing figures, those skilledin the art will understand the concepts of the invention and willrecognize applications of these concepts not particularly addressedherein. It should be understood that these concepts and applicationsfall within the scope of the disclosure and the accompanying claims.

The present invention relates to a polychronous wave propagation systemthat is based on relative timing between two or more propagated wavesthrough a wave propagation medium. The relative timing may be associatedwith interference patterns of energy between the propagated waves.Operational behavior of the polychronous wave propagation system isbased on the relative timing of the propagated waves and distancesbetween initiators that transmit the propagated waves and respondersthat receive the propagated waves. The operational behavior may includearithmetical computations, memory storage, Boolean functions,frequency-based computations, or the like. The arithmetical computationsmay include additions, subtractions, multiplications, divisions, or anycombination thereof. Navigation systems may use interference patternsand transponders in a variety of ways. However, embodiments of thepresent disclosure differ from such systems by using a polychronous wavepropagation system that is analogous to polychronous activity in abrain. Further, the polychronous wave propagation system may store theresults of computations using reverberating systems.

The polychronous wave propagation system relies on time delays betweenthe propagated waves that result from propagation velocities of thepropagated waves through the wave propagation medium. By incorporatingthe time delays into the polychromous wave propagation system,operational capacity may be greatly enhanced. Functionally, thepropagated waves may be used to actuate certain events after certaintime delays. As such, the operational behavior may be based on theactuation of the certain events. Computations may be carried out byutilizing interference patterns of activity associated with preciselyplaced transponders. Reverberating arrays may be used to store resultsof specific computational tasks.

In one embodiment of the present invention, the propagated wave may be apulsed wave lasting just a few cycles. Further, a responder and aninitiator may be combined to form a transponder, which may receive twoor more propagated waves and initiate a response transmission of apropagated wave based on the relative timing between the receivedpropagated waves. In an exemplary embodiment of the present invention,the response transmission is initiated based on simultaneous receptionof the received propagated waves. In general, the relative timing may beassociated with interference patterns of energy between the receivedpropagated waves. The transponder may include a controllable oscillatorblock, which may be controlled based on the responder section of thetransponder and may provide a controlled signal upon which the responsetransmission is based. In one embodiment of the controllable oscillatorblock, the controllable oscillator block oscillates at a controllablefrequency, which may be based on frequencies of received pulsedpropagated waves. The initiated transmission may be a pulsed propagatedwave that may be at the frequency of the controllable oscillator block.

FIGS. 1A, 1B, 1C, and 1D may represent snapshots in time of a first wavepropagation medium 10 at zero seconds elapsed time, one second elapsedtime, two seconds elapsed time, and three seconds elapsed time,respectively. Behaviors of a polychronous wave propagation systeminvolve propagating waves through the first wave propagation medium 10.To illustrate the behaviors, an analogy of waves on a lake will be used.A smooth surface of a lake may function as the first wave propagationmedium 10. A first bobber, which may serve as a first initiator 12, mayfloat on the surface of the lake. At an elapsed time of zero seconds,the first initiator 12 is about to generate a first wave pulse 14 (notshown) as illustrated in FIG. 1A. In the waves on a lake analogy, thefirst bobber may generate a wave, which may be similar to dropping anobject, such as a rock, into the water. The first wave pulse 14 radiatesoutward in a radial manner from the first initiator 12, as illustratedin FIG. 1B, which is a snapshot at an elapsed time of one second. In thewaves on a lake analogy, the first wave pulse 14 may be represented as aleading crest of the wave, which has a maximum height. The leading crestof the wave may be followed by a second crest 16 of lesser height and athird crest 18 of even lesser height. FIG. 1C shows the outwardpropagation of the wave at an elapsed time of two seconds and FIG. 1Dshows the outward propagation of the wave at an elapsed time of threeseconds.

FIGS. 2A through 2H show a polychronous wave propagation system 20, inwhich asynchronous waves that have timing relationships to one anotherare transmitted and propagate through the first wave propagation medium10. FIGS. 2A, 2B, 2C, 2D, 2E, 2F, 2G, and 2H may represent snapshots intime of the first wave propagation medium 10 at zero seconds elapsedtime, two seconds elapsed time, six seconds elapsed time, eight secondselapsed time, eleven seconds elapsed time, fourteen seconds elapsedtime, eighteen seconds elapsed time, and twenty-two seconds elapsedtime, respectively.

The polychronous wave propagation system 20 includes the first initiator12, which is positioned along an edge of the first wave propagationmedium 10, a second initiator 22, which is positioned along another edgeof the first wave propagation medium 10, a first transponder 24, asecond transponder 26, a third transponder 28, and a fourth transponder30. The first and the second initiators 12, 22, the first, the second,the third, and the fourth transponders 24, 26, 28, 30 may be carefullylocated with respect to one another in fixed positions in the first wavepropagation medium 10 to produce desired behaviors in the polychronouswave propagation system 20. The first and the second initiators 12, 22are only capable of transmitting waves, whereas the first, the second,the third, and the fourth transponders 24, 26, 28, 30 are capable ofboth receiving propagated waves and transmitting waves. In a firstexemplary embodiment of the present invention, any of the first, thesecond, the third, and the fourth transponders 24, 26, 28, 30 transmit awave in response to receiving two wave pulses simultaneously.

In the waves on a lake analogy, the first and the second initiators 12,22 may be represented as first and second bobbers, respectively, thatare only capable of creating waves in the smooth lake surface. Thefirst, the second, the third, and the fourth transponders 24, 26, 28, 30may be represented as third, fourth, fifth and sixth bobbers,respectively, that are floating on the smooth lake surface waiting forwaves to come along. When any of the third, the fourth, the fifth andthe sixth bobbers encounters two simultaneous leading wave crests, whichis representative of receiving two wave pulses simultaneously, thesimultaneous leading wave crests add to one another, thereby creating awave height that may be about double the height of an individual leadingwave crest. In the first exemplary embodiment of the present invention,the third, the fourth, the fifth and the sixth bobbers are onlyresponsive to about double height wavecrests, thereby transmitting awave in response.

At an elapsed time of zero seconds, the first initiator 12 is about togenerate a first wave pulse 14 (not shown) as illustrated in FIG. 2A.The first wave pulse 14 propagates outward in a radial manner from thefirst initiator 12, as illustrated in FIG. 2B, which is a snapshot at anelapsed time of two seconds. At an elapsed time of six seconds, thesecond initiator 22 is about to generate a second wave pulse 32 (notshown) and the first wave pulse 14 continues to propagate outward in aradial manner from the first initiator 12 as illustrated in FIG. 2C. Thesecond wave pulse 32 propagates outward in a radial manner from thesecond initiator 22 and the first wave pulse 14 continues to propagateoutward in a radial manner from the first initiator 12 as illustrated inFIG. 2D, which is a snapshot at an elapsed time of eight seconds.Therefore, the second wave pulse 32 lags the first wave pulse 14 by sixseconds. At an elapsed time of eleven seconds, the first wave pulse 14reaches the first transponder 24; however, since the first transponder24 is only responsive to simultaneous reception of two wave pulses, thefirst transponder 24 doesn't respond, and both the first and the secondwave pulses 14, 32 continue to propagate outward in a radial manner fromthe first and the second initiators 12, 22 as illustrated in FIG. 2E.

At an elapsed time of fourteen seconds, the first wave pulse 14 reachesthe second transponder 26; however, since the second transponder 26 isonly responsive to simultaneous reception of two wave pulses, the secondtransponder 26 does not respond, and both the first and the second wavepulses 14, 32 continue to propagate outward in a radial manner from thefirst and the second initiators 12, 22, respectively, as illustrated inFIG. 2F. At an elapsed time of eighteen seconds, both the first and thesecond wave pulses 14, 32 reach the third transponder 28, and since thethird transponder 28 is responsive to simultaneous reception of two wavepulses, the third transponder 28 is about to respond by transmitting athird wave pulse 34 (not shown) as illustrated in FIG. 2G. At an elapsedtime of twenty-two seconds, the first, the second, and the third wavepulses 14, 32, 34 propagate outward in a radial manner from the firstand the second initiators 12, 22, and the third transponder 28,respectively, as illustrated in FIG. 2H.

The operational behavior of the polychronous wave propagation system 20is demonstrated by the specific outcome illustrated in FIGS. 2A through2H, which is an initiation of transmission of the third wave pulse 34from the third transponder 28 at an elapsed time of eighteen seconds andno response from any of the first, the second, and the fourthtransponders 24, 26, 30 is due to the timing relationship between thefirst and the second wave pulses 14, 32 and the precise physicallocations of the first, the second, the third, and the fourthtransponders 24, 26, 28, 30, and the first and the second initiators 12,22 with respect to one another. Any other timing relationship, any otherphysical arrangement, or both may have significantly affected theoutcome.

The first and the second initiators 12, 22 transmitted waves havingencoded information. In the waves on a lake analogy, the height of theleading wave crests is the encoded information. Responders in the first,the second, the third, and the fourth transponders 24, 26, 28, 30receive and detect information based on timing between reception of theencoded information from the first and the second initiators 12, 22. Inthe waves on a lake analogy, the transponders are only responsive todouble height wave crests; therefore, the detected information is basedon simultaneous reception of leading wave crests that produces a doubleheight wave.

FIG. 3 is a three-dimensional graph showing space-time relationships ofthe polychronous wave propagation system 20 illustrated in FIGS. 2Athrough 2H. The purpose of the three-dimensional graph is tosimultaneously illustrate relationships between the physical locationsof the devices and the timing relationships between the wave pulses tohelp understand the operational behavior of the polychronous wavepropagation system 20. The three-dimensional graph includes X, Y, and Taxes. The first wave propagation medium 10 is overlaid on the X-Y planewith the origin of the X-Y plane located at the location of the firstinitiator 12. The T-axis represents elapsed time. The origin of the X,Y, and T axes represents an elapsed time of zero seconds. As one movesin a positive direction along the T-axis, elapsed time increases.Space-time cones may represent the propagation of waves through thefirst wave propagation medium 10. A first space-time cone 36 representspropagation of the first wave pulse 14 (not shown) and a secondspace-time cone 38 represents propagation of the second wave pulse 32(not shown).

At the elapsed time of zero seconds, the first initiator 12 is about totransmit the first wave pulse 14; therefore, the vertex of the firstspace-time cone 36 is located at the origin of the X, Y, and T axes. Thefirst space-time cone 36 includes a first ring 40, a second ring 42, anda third ring 44, which are representative of the first wave pulse 14 atelapsed times of six seconds, twelve seconds, and eighteen seconds,respectively. At the elapsed time of six seconds, the second initiator22 is about to transmit the second wave pulse 32; therefore, a vertex 46of the second space-time cone 38 is located at the location of thesecond initiator 22 in the X-Y plane and at six seconds along theT-axis. The distance from the second initiator 22 in the X-Y plane andthe vertex 46 of the second space-time cone 38 represents a delay 48between the start of the transmission of the first wave pulse 14 and thestart of the transmission of the second wave pulse 32 (not shown).

The second space-time cone 38 includes a fourth ring 50 and a fifth ring52, which are representative of the second wave pulse 32 at elapsedtimes of twelve seconds and eighteen seconds, respectively. At anelapsed time of twelve seconds, the first wave pulse 14 and the secondwave pulse 32 intersect each other for the first time; therefore, thesecond ring 42 and the fourth ring 50 intersect each other at a firstintersection point 54. At an elapsed time of eighteen seconds, the firstwave pulse 14 and the second wave pulse 32 intersect each other at thelocation of the third transponder 28; therefore, the third ring 44 andthe fifth ring 52 intersect each other at a second intersection point56.

By taking a first projection 58 of the first intersection point 54 ontothe X-Y plane parallel to the T-axis, a point on a first parabola 62 islocated. By taking a second projection 60 of the second intersectionpoint 56 onto the X-Y plane parallel to the T-axis, the thirdtransponder 28 is located, which is also a point that falls on the firstparabola 62. If all of the intersection points between the first and thesecond space-time cones 36, 38 are projected onto the X-Y plane, allwould fall on the first parabola 62, which is called a parabola ofintersection. Any of the first, the second, the third, and the fourthtransponders 24, 26, 28, 30 located on the parabola of intersectionwould simultaneously receive the first and the second wave pulses 14,32, but at different elapsed times. Different locations of the firstinitiator 12, the second initiator 22, or both, different delays betweenthe first and the second wave pulses 14, 32, or any combination thereof,would normally have different parabolas of intersection. Therefore,identifying the parabolas of intersection may be very important inidentifying operational behaviors of the polychronous wave propagationsystem 20 and designing the polychronous wave propagation system 20. Aswill be apparent to one of ordinary skill in the art, the parabola ofintersection may be defined by the actual propagation of the first wavepulse 14 and the second wave pulse 32 through the first wave propagationmedium 10, and is defined by the timing relationship between the firstwave pulse 14 and the second wave pulse 32. Small variations in thefirst wave pulse 14 and the second wave pulse 32 as they propagatethrough the first wave propagation medium 10 may cause the parabola ofintersection to not be in the shape of a perfect parabola. In thisregard, the term “parabola of intersection” may refer to a shape formedin the first wave propagation medium 10 based on the timing oftransmission between the first wave pulse 14 and the second wave pulse32. As such, the parabola of intersection is generally in the shape of aparabola, but may not be in the shape of a perfect parabola.

FIG. 4 shows details of the polychronous wave propagation system 20illustrated in FIGS. 2A through 2H, according to one embodiment of thepolychronous wave propagation system 20. The polychronous wavepropagation system 20 includes the first and the second initiators 12,22, the first, the second, the third, and the fourth transponders 24,26, 28, 30, a control system 64, information aggregation circuitry 66,information distribution circuitry 68, and initialization distributioncircuitry 70. The control system 64, which may include controlcircuitry, provides an aggregated initialization control signal ICS tothe initialization distribution circuitry 70, which provides a firstinitialization control signal ICS₁ to the first transponder 24, a secondinitialization control signal ICS₂ to the second transponder 26, a thirdinitialization control signal ICS₃ to the third transponder 28, a fourthinitialization control signal ICS₄ to the fourth transponder 30, a fifthinitialization control signal ICS₅ to the first initiator 12, and asixth initialization control signal ICS₆ to the second initiator 22based on the aggregated initialization control signal ICS. The first,the second, the third, the fourth, the fifth, and the sixthinitialization control signals ICS₁, ICS₂, ICS₃, ICS₄, ICS₅, ICS₆ mayprovide initialization information from the control system 64 to thefirst, the second, the third, and the fourth transponders 24, 26, 28,30, and the first and the second initiators 12, 22, respectively, whichmay use the initialization information to prepare the polychronous wavepropagation system 20 for initial use, for use between operatingsessions, for use during operating sessions, or the like.

The control system 64 provides aggregated initiator control informationINCI to the information distribution circuitry 68, which provides firstinitiator control information INCI₁ to the first initiator 12 and secondinitiator control information INCI₂ to the second initiator 22 based onthe aggregated initiator control information INCI. The first and thesecond initiator control information INCI₁, INCI₂ may provideoperational control information from the control system 64 to the firstand the second initiators 12, 22, respectively, which may use theoperational control information to control operations of the first andthe second initiators 12, 22. Such control operations may includeinitiating transmission of waves or wave pulses, selecting wave pulsedurations, establishing timing relationships between the first and thesecond initiators 12, 22, or the like.

The control system 64 receives aggregated responder information RI fromthe information aggregation circuitry 66, which receives first responderinformation RI₁ from the first transponder 24. In general, theaggregated responder information RI is the mechanism for conveyinginformation from one or more responders in the polychronous wavepropagation system 20 to the control system 64, which may convey theinformation to entities outside of the polychronous wave propagationsystem 20.

FIG. 5 shows details of the first, the second, the third, and the fourthtransponders 24, 26, 28, 30 illustrated in FIG. 4, according to oneembodiment of the polychronous wave propagation system 20. The firsttransponder 24 includes a first responder 72 and a third initiator 74,the second transponder 26 includes a second responder 76 and a fourthinitiator 78, the third transponder 28 includes a third responder 80 anda fifth initiator 82, and the fourth transponder 30 includes a fourthresponder 84 and a sixth initiator 86.

FIG. 6 shows details of the first transponder 24 illustrated in FIG. 5,according to one embodiment of the first transponder 24. As mentionedabove, the first transponder 24 includes the first responder 72 and thethird initiator 74. The first responder 72 provides the first responderinformation RI₁ to the control system 64 (not shown), the thirdinitiator 74 receives the first initialization control signal ICS₁ fromthe control system 64 (not shown), and the first responder 72 provides afirst forcing signal FS₁ to the third initiator 74. Since the firsttransponder 24 transmits a wave in response to receiving two wave pulsessimultaneously, when the first responder 72 receives two wave pulsessimultaneously, the first responder 72 drives the first forcing signalFS₁ to cause the third initiator 74 to transmit a response wave.Additionally, the first responder 72 may indicate to the control system64 that two wave pulses were received simultaneously by providing theappropriate first responder information RI₁.

FIG. 7 shows details of the first responder 72 and the third initiator74 illustrated in FIG. 6, according to one embodiment of the firsttransponder 24. The first responder 72 includes a first receivetransducer 88 and first receive circuitry 90. The third initiator 74includes first transmit circuitry 92 and a first transmit transducer 94.The first receive circuitry 90 provides the first responder informationRI₁ to the control system 64 (not shown), the first transmit circuitry92 receives the first initialization control signal ICS₁ from thecontrol system 64 (not shown), the first receive transducer 88 providesfirst detected information DI₁ to the first receive circuitry 90, whichprovides the first forcing signal FS₁ to the first transmit circuitry92, and the first transmit circuitry 92 provides a first transmit signalTS₁ to the first transmit transducer 94. The first transmit signal TS₁includes appropriate response information to be transmitted.

The first wave propagation medium 10 may include air; vacuum; a liquid,such as water; a solid material, such as insulating material;semiconductor material; a surface of a semiconductor layer; multiplelayers of semiconductor material; conducting material; magneticmaterial; magnetic films; ferromagnetic material; ferromagnetic materialhaving an array of spin-torque nano-oscillators; ferrimagnetic material;any other material; an interface between any two of the aforementioned;any combination thereof; or the like. The first wave propagation medium10 may be three dimensional, planar, multiple planes, circular, square,rectangular, triangular, spherical, cubic, cylindrical, cone-shaped,dodecahedronal, any size, any shape, any combination thereof, or thelike. The first wave propagation medium 10 may form a micro-structure ora nano-structure. As such, the first wave propagation medium 10 may berelatively small. The temporal resolution of the polychronous wavepropagation system 20 may be in the range of gigahertz and the spacescale of the polychronous wave propagation system 20 may be in the rangeof tens of nanometers.

In a first embodiment of the first wave propagation medium 10, a longestdimension of the first wave propagation medium 10 is less than about onemeter. In a second embodiment of the first wave propagation medium 10,the longest dimension of the first wave propagation medium 10 is lessthan about one centimeter. In a third embodiment of the first wavepropagation medium 10, the longest dimension of the first wavepropagation medium 10 is less than about 100 micrometers. In a fourthembodiment of the first wave propagation medium 10, the longestdimension of the first wave propagation medium 10 is less than about onemicrometer. In a fifth embodiment of the first wave propagation medium10, the longest dimension of the first wave propagation medium 10 isless than about 500 nanometers.

The propagating waves may be surface waves on a lake, waves on thesurface of other fluids, acoustic waves on a semiconductor substrate,waves in a gaseous or liquid medium, such as Mercury, electro-magneticwaves, such as light, radio frequency signals, or x-rays, shock-waves inelastic materials, solitons in super-conducting circuits, quantum wavepackets in arrays of spin-torque oscillators, such as nano-oscillatorsor ferromagnetic oscillators, or the like. The waves may be isolatedradial activity waves or other types of waves. The waves may be capableof passing through one another without significant interaction with oneanother, with some interaction with one another, such as imparting aphase-shift, or with significant interaction with one another, such asreinforcement or attenuation. In general, a propagating wave isassociated with a packet of energy that can actuate something in thefirst wave propagation medium 10 after a certain wave propagation timedelay. The propagating waves may be represented by the wave equation orby Schrodinger's equation, which may be indicative of certain wave-likecharacteristics and certain particle-like characteristics.

The wave equation is a second-order linear partial differential equationof waves and is a prototypical example of a hyberbolic partialdifferential equation as shown below.

$\frac{\partial^{2}u}{\partial t^{2}} = {c^{2}{\nabla^{2}u}}$

where ∇² is the Laplacian and where c is a fixed constant equal to thepropagation speed of the waves characterized by the wave equation. Inits simplest form, the wave equation refers to a scalar function u=u(x₁,x₂, . . . , x_(n), t) that satisfies the wave equation. The general formof Schrodinger's equation for a general quantum system is shown below.

${{\hslash}\frac{\partial}{\partial t}\Psi} = {\hat{H}\; \Psi}$

where ψ is the wave function, which is the probability amplitude fordifferent configurations of the system at different times,

${\hslash}\frac{\partial}{\partial t}$

is the energy operator (i is the imaginary unit and h is the reducedPlanck's constant), and

Ĥ is the Hamiltonian operator.

Since the propagating waves may have such a wide range ofcharacteristics, the first receive transducer 88 may be needed toproperly receive and extract the first detected information DI₁ fromreceived propagating waves. Similarly, the first transmit transducer 94properly transmits the response waves using the appropriate responseinformation contained in the first transmit signal TS₁.

FIG. 8 shows details of the first transmit circuitry 92 illustrated inFIG. 7, according to one embodiment of the first transmit circuitry 92.As mentioned above, the third initiator 74 includes the first transmitcircuitry 92 and the first transmit transducer 94. The first transmitcircuitry 92 includes a first controllable oscillator block 96 andsignal gating and delay circuitry 98. The first controllable oscillatorblock 96 receives the first initialization control signal ICS₁ from thecontrol system 64 (not shown) and the first forcing signal FS₁ from thefirst receive circuitry 90 (not shown). The first controllableoscillator block 96 provides a first output signal OS₁, which isrepresentative of first output information, to the signal gating anddelay circuitry 98, which provides the first transmit signal TS₁ to thefirst transmit transducer 94 based on the first output information. Ingeneral, the first controllable oscillator block 96 is a device that canreceive signals, process the received signals in ways that may becomplex, and emit signals based on the processing of the receivedsignals. The first controllable oscillator block 96 may expressself-sustained oscillatory output or may be excitable, depending oncontrol signals provided to the first controllable oscillator block 96.A mechanical example of the first controllable oscillator block 96 is amechanical pendulum having a torque applied to the supporting point.When at rest, a few precisely timed pushes may cause the pendulum toexhibit a complete rotation, or if the forcing torque is sufficientlylarge, the pendulum might exhibit ongoing oscillations. In oneembodiment of the present invention, the first controllable oscillatorblock 96 may include a voltage controlled oscillator neuron (VCON)oscillator. The first output information may include a first frequencyand the first detected information DI₁ may include a second frequency.The first receive circuitry 90 may provide the first forcing signal FS₁to control the first controllable oscillator block 96, such that thefirst frequency is controlled to be about equal to the second frequency.

In alternate embodiments of the present invention, controllableoscillator blocks, such as the first controllable oscillator block 96,may be implemented using phase-locked loops, frequency-locked loops,memristors, spin-torque nano-oscillators, voltage controlled oscillatorblocks, current controlled oscillator blocks, frequency controlledoscillator blocks, Josephson junctions, single-electron transistors,certain chemical reaction arrays, any combination thereof, or the like.

FIG. 9 shows details of the polychronous wave propagation system 20,according to an alternate embodiment of the polychronous wavepropagation system 20. The polychronous wave propagation system 20includes a system array 100, which includes a first responder array 102,a first transponder array 104, and a first initiator array 106.Additionally, the polychronous wave propagation system 20 includes thecontrol system 64, the information aggregation circuitry 66, theinformation distribution circuitry 68, and the initializationdistribution circuitry 70. The control system 64 provides an aggregatedinitialization control signal ICS to the initialization distributioncircuitry 70, which provides a first initialization control signal ICS₁,a second initialization control signal ICS₂, up to and including a Pthinitialization control signal ICS_(P) to the system array 100 based onthe aggregated initialization control signal ICS. The first, the second,up to and including the Pth initialization control signals ICS₁, ICS₂,ICS_(P) may provide initialization information from the control system64 to the system array 100, which may use the initialization informationto prepare the polychronous wave propagation system 20 for initial use,for use between operating sessions, for use during operating sessions,or the like.

The control system 64 provides aggregated initiator control informationINCI to the information distribution circuitry 68, which provides firstinitiator control information INCI₁, second initiator controlinformation INCI₂, up to and including Nth initiator control informationINCI_(N) to the first initiator array 106 based on the aggregatedinitiator control information INCI. The first, the second, up to andincluding the Nth initiator control information INCI₁, INCI₂, INCI_(N)may provide operational control information from the control system 64to the first initiator array 106, which may use the operational controlinformation to control operations of the first initiator array 106. Suchoperations may include initiating transmission of waves or wave pulses,selecting wave pulse durations, establishing timing relationshipsbetween elements in the first initiator array 106, or the like.

The control system 64 receives aggregated responder information RI fromthe information aggregation circuitry 66, which receives first responderinformation RI₁, second responder information RI₂, up to and includingMth responder information RI_(M) from the first responder array 102. Ingeneral, the aggregated responder information RI is the mechanism forconveying information from one or more responders in the polychronouswave propagation system 20 to the control system 64, which may conveythe information to entities outside of the polychronous wave propagationsystem 20.

FIG. 10 shows details of the system array 100 of initiators, responders,and transponders disposed in the first wave propagation medium 10. Thesystem array 100 includes the first responder array 102, the firsttransponder array 104, and the first initiator array 106. The firstresponder array 102 includes a column of responders disposed along anedge of the first wave propagation medium 10. The responders are onlycapable of receiving waves. The first initiator array 106 includes acolumn of initiators disposed along an opposite edge of the first wavepropagation medium 10. The initiators are only capable of transmittingwaves. The first transponder array 104 includes multiple columns oftransponders disposed between the first responder array 102 and thefirst initiator array 106. The transponders are capable of bothreceiving propagated waves and transmitting waves. Each of thetransponders in the first transponder array 104 has a responder andinitiator pair.

In one embodiment of the present invention, each of at least twoinitiators in the first initiator array 106, in the first transpondersarray 104, or both are used to provide stimulus into the polychronouswave propagation system 20 by transmitting waves having encodedinformation through the first wave propagation medium 10 to at least oneresponder in the first transponder array 104, the first responder array102, or both. The transmitted waves may be received mainly by nearestneighbors in the first transponder array 104 or may pass by the nearestneighbors to other responders. The timing of transmission of thetransmitted waves having the encoded information is based on theaggregated initiator control information INCI, which is provided by thecontrol system 64. Each of at least two responders in the firsttransponder array 104, the first responder array 102, or both arecapable of receiving transmitted waves from at least two initiators inthe first transponder array 104, the first initiator array 106, or both,and detecting information based on timing between reception of theencoded information in each of the received waves. In an exemplaryembodiment of the present invention, the detected information is basedon simultaneous reception of the encoded information in each of thereceived waves.

In the first transponder array 104, each initiator in at least oneresponder and initiator pair transmits response waves in response to thedetected information associated with the waves that were received by theresponder in the responder and initiator repair. The timing oftransmission of the encoded information in the response waves is basedon the detected information.

Aggregated responder information RI is provided to the control system 64based on the detected information from each of at least one responder inthe first responder array 102. Operational behavior of the polychronouswave propagation system 20 is based on the detected information providedby the responders and distances between each active initiator and eachactive responder in the polychronous wave propagation system 20.

In alternate embodiments of the polychronous wave propagation system 20,initiator control information based on the aggregated initiator controlinformation INCI may be provided to initiators in the first transponderarray 104, aggregated responder information RI may be provided to thecontrol system 64 based on the detected information from each of atleast one responder in the first transponder array 104. The firsttransponder array 104 includes at least one responder and initiatorpair, the first responder array 102 includes any number of responders,which may include zero, and the first initiator array 106 includes anynumber of initiators, which may include zero.

In other embodiments of the present invention, the polychronous wavepropagation system 20 includes any number of initiators and any numberof responders. Further, any number of initiators may be combined with anequal number of responders to form transponders.

In one embodiment of the present invention, the transmitted waves fromat least one of the initiators in the first initiator array 106, thefirst transponder array 104, or both, are amplitude modulated waves andthe encoded information associated with the transmitted waves includesamplitude modulation information. Amplitude modulation associated withthe amplitude modulated waves may include ON-OFF keying, such that theamplitude modulated waves are pulsed waves.

In one embodiment of the present invention, the transmitted waves fromat least one of the initiators in the first initiator array 106, thefirst transponder array 104, or both have a first frequency and theencoded information associated with the transmitted waves includes firstfrequency information.

In one embodiment of the present invention, the transmitted waves fromat least one of the initiators in the first initiator array 106, thefirst transponder array 104, or both are TRUE reference signals and theencoded information associated with the transmitted waves includes TRUEreference information. Further, the transmitted waves from at leastanother of the initiators in the first initiator array 106, the firsttransponder array 104, or both are FALSE reference signals and theencoded information associated with the transmitted waves includes FALSEreference information.

In one embodiment of the present invention, operational behavior of thepolychronous wave propagation system 20 is representative of at leastone Boolean function. In an alternate embodiment of the presentinvention, operational behavior of the polychronous wave propagationsystem 20 includes frequency detection of the transmitted wavesassociated with each of at least one of the initiators in the firstinitiator array 106, the first transponder array 104, or both. In anadditional embodiment of the present invention, operational behavior ofthe polychronous wave propagation system 20 is representative of atleast one reverberating memory cell. In another embodiment of thepresent invention, operational behavior of the polychronous wavepropagation system 20 is representative of at least one reverberatingmemory doublet cell.

In one embodiment of the present invention, the first wave propagationmedium 10 includes a ferromagnetic thin film of magnetic material, whichhas an array of spin-torque nano-oscillators, such that eachnano-oscillator has a magnetization vector. In the absence of anyexternal influences, the magnetization vectors may be about aligned toone another. However, in the presence of an external magnetic field thatis misaligned with respect to the magnetization vectors at an edge ofthe array, some of the nano-oscillators at the edge of the array mayre-align themselves with the external magnetic field. Then,nano-oscillators that are adjacent to the edge nano-oscillators mayre-align themselves, with the edge nano-oscillators. Next, adjacentoscillators to re-aligned oscillators may re-align themselves with there-aligned oscillators, and so on. The re-alignment of nano-oscillatorsmay propagate through the array as waves through a wave propagationmedium, which is the array of spin-torque nano-oscillators. The wavepropagation is the propagation of a spin wave by spin-torque. When theexternal magnetic field is removed, the edge nano-oscillators may revertback to being aligned with their corresponding magnetization vectors.The nano-oscillators that are adjacent to the edge nano-oscillators mayrevert back to being aligned with their corresponding magnetizationvectors, and so on.

In a polychronous wave propagation system 20 that includes theferromagnetic thin film of magnetic material having the array ofspin-torque nano-oscillators as the first wave propagation medium 10,initiators may have the capability of applying an external magneticfield to some of the spin-torque nano-oscillators in the first wavepropagation medium 10. Further, responders may include spin-torquenano-oscillators that are responsive to the spin wave and may include adetection device to detect the re-alignment of the spin-torquenano-oscillators.

Examples of different applications using the polychronous wavepropagation system 20 are presented in different embodiments of thepresent invention. In a first example, Boolean functionality using thepolychronous wave propagation system 20 is presented. Boolean functionsare the fundamental building blocks of most digital computers. BasicBoolean building blocks are binary in nature, such that any given nodemay be in a TRUE state or a FALSE state. In one embodiment of thepresent invention, the polychronous wave propagation system 20 is usedto implement a NOR gate. Combinations of NOR gates may be used toprovide any Boolean function.

FIGS. 11A through 11F and 12A through 12G illustrate differentapproaches to defining TRUE and FALSE states, which are used by thepolychronous wave propagation system 20 to provide Booleanfunctionality. In a first approach, FIGS. 11A and 11B illustrate areceived signal indicating a TRUE state and a FALSE state, respectively.In the TRUE received signal, a first pulse 108 is indicative of the TRUEstate and in the FALSE received signal, an absence of pulses isindicative of the FALSE state. The first approach is not very practicalsince a silent transponder could be interpreted as constantlytransmitting FALSE signals.

In a second approach, FIGS. 11C and 11D illustrate a received signalindicating the TRUE state and the FALSE state, respectively. In the TRUEreceived signal, the presence of the first pulse 108 and a second pulse110 is indicative of the TRUE state and in the FALSE received signal,the presence of only the first pulse 108 is indicative of the FALSEstate.

In a third approach, FIGS. 11E and 11F illustrate a received signalindicating the TRUE state and the FALSE state, respectively. In the TRUEreceived signal, the presence of the first pulse 108 and the secondpulse 110 separated by a first inter-pulse interval is indicative of theTRUE state and in the FALSE received signal, the presence of the firstpulse 108 and the second pulse 110 separated by a second inter-pulseinterval is indicative of the FALSE state, such that the firstinter-pulse interval is not equal to the second inter-pulse interval toallow differentiation between the TRUE state and the FALSE state.

In a fourth approach, FIGS. 12A and 12B illustrate a received signalindicating the TRUE state and the FALSE state, respectively. FIG. 12Cillustrates a timing reference signal having equally spaced timingreference pulses 112. In the TRUE received signal, the presence of thefirst pulse 108 separated from one of the timing reference pulses 112 bya first interval 114 is indicative of the TRUE state and in the FALSEreceived signal, the presence of the second pulse 110 separated from oneof the timing reference pulses 112 by a second interval 116 isindicative of the FALSE state, such that the first interval 114 is notequal to the second interval 116 to allow differentiation between theTRUE state and the FALSE state. Further, the first pulse 108 may becalled a TRUE pulse and the second pulse 110 may be called a FALSEpulse, such that the first pulse 108 and the second pulse 110 aredifferentiated from one another. For example, pulse widths of the firstand the second pulses 108, 110 may be different from one another, orcarrier frequencies of the first and the second pulses 108, 110 may bedifferent from one another.

In a fifth approach, FIGS. 12D and 12E illustrate a received signalindicating the TRUE state and the FALSE state, respectively. FIG. 12Fillustrates a TRUE timing reference signal having equally spaced TRUEtiming reference pulses 118 and FIG. 12G illustrates a FALSE timingreference signal having equally spaced FALSE timing reference pulses120. In the TRUE received signal, the presence of the first pulse 108received about simultaneously with one of the TRUE timing referencepulses 118 is indicative of the TRUE state and in the FALSE receivedsignal, the presence of the second pulse 110 received aboutsimultaneously with one of the FALSE timing reference pulses 120 isindicative of the FALSE state. As mentioned above, the first pulse 108may be called a TRUE pulse and the second pulse 110 may be called aFALSE pulse, such that the first pulse 108 and the second pulse 110 aredifferentiated from one another. For example, pulse widths of the firstand the second pulses 108, 110 may be different from one another, orcarrier frequencies of the first and the second pulses 108, 110 may bedifferent from one another. Similarly, TRUE timing reference pulses 118may be differentiated from FALSE timing reference pulses 120. Forexample, pulse widths of the TRUE and the FALSE timing reference pulses118, 120 may be different from one another, or carrier frequencies ofthe TRUE and the FALSE timing reference pulses 118, 120 may be differentfrom one another.

The fifth approach presented above is used by the polychronous wavepropagation system 20 in the first example to provide a NOR gate. TheNOR gate has an A input, a B input, and an inverting output. When eitheror both of the A input or the B input is TRUE, the inverting output isFALSE. When both the A input and the B input are FALSE, the invertingoutput is TRUE. The NOR gate is implemented as an OR gate and aninverter. The OR gate has the A input, the B input, and a non-invertingoutput. The inverter takes the non-inverting output and provides theinverting output of the NOR gate based on inverting the state of thenon-inverting output. In the OR gate, when either or both of the A inputor the B input is TRUE, the non-inverting output is TRUE. When both theA input and the B input are FALSE, the inverting output is FALSE.

FIGS. 13A through 16E illustrate behaviors of the polychronous wavepropagation system 20 in which asynchronous waves that have timingrelationships to one another are transmitted and propagate through thefirst wave propagation medium 10. The polychronous wave propagationsystem 20 includes the first initiator 12, which is positioned along anedge of the first wave propagation medium 10 and provides the FALSEtiming reference signal, the second initiator 22, which is positionedalong another edge of the first wave propagation medium 10 and providesthe TRUE timing reference signal, the first transponder 24, whichprovides the non-inverting output, the second transponder 26, whichprovides the A input, the third transponder 28, which provides the Binput, the fourth transponder 30, which is used to provide theappropriate behavior, and a fifth transponder 122, which is used toprovide the inverting output. The first and the second initiators 12,22, the first, the second, the third, the fourth, and the fifthtransponders 24, 26, 28, 30, 122 may be carefully located with respectto one another to produce desired behaviors in the polychronous wavepropagation system 20.

The second and the third transponders 26, 28 are only responsive totiming pulses that match the states of the A input and the B input,respectively. For example, if the A input is in a TRUE state, the secondtransponder 26 will ignore FALSE timing reference pulses 120 but beresponsive to TRUE timing reference pulses 118. Likewise, if the A inputis in a FALSE state, the second transponder 26 will ignore TRUE timingreference pulses 118 but be responsive to FALSE timing reference pulses120. The B input and third transponder 28 behave in a similar manner.The first transponder 24, which provides the non-inverting output is notresponsive to single pulses, but is responsive to simultaneous pulseshaving the same state by transmitting a pulse with the same state. Forexample if the first transponder 24 receives two or more TRUE pulsessimultaneously, the first transponder 24 responds by transmitting a TRUEpulse. Similarly, if the first transponder 24 receives two or more FALSEpulses simultaneously, the first transponder 24 responds by transmittinga FALSE pulse.

The fourth transponder 30 is only responsive to two simultaneous FALSEpulses, such that if the fourth transponder 30 receives two or moreFALSE pulses simultaneously, the fourth transponder 30 responds bytransmitting a FALSE pulse. The fifth transponder 122, which providesthe inverting output is not responsive to single pulses, but isresponsive to simultaneous pulses having the same state by transmittinga pulse with the opposite state. For example if the fifth transponder122 receives two or more TRUE pulses simultaneously, the fifthtransponder 122 responds by transmitting a FALSE pulse. Similarly, ifthe fifth transponder 122 receives two or more FALSE pulsessimultaneously, the fifth transponder 122 responds by transmitting aTRUE pulse.

FIGS. 13A through 13E illustrate behavior of the NOR gate using thepolychronous wave propagation system 20 with both A and B inputs of theNOR gate being TRUE; therefore, the inverting output of the NOR gateshould be FALSE. The states of the A and B inputs may have been driveninto these states by other initiators (not shown), by circuitry drivenby the aggregated initiator control information INCI from the controlsystem 64, or both. FIGS. 13A, 13B, 13C, 13D, and 13E may representsnapshots in time of the first wave propagation medium 10 at two timeunits elapsed time, nine time units elapsed time, twelve time unitselapsed time, eighteen time units elapsed time, and twenty-four timeunits elapsed time, respectively.

FIG. 13A, which represents a snapshot at two time units elapsed time,shows a TRUE timing reference pulse 118, which was transmitted from thesecond initiator 22 about two time units previously and a FALSE timingreference pulse 120, which was transmitted from the first initiator 12about six time units previously; therefore, the TRUE timing referencepulses 118 lag respective FALSE timing reference pulses 120 by aboutfour time units. FIG. 13B, which represents a snapshot at nine timeunits elapsed time, shows the TRUE timing reference pulse 118 reachingthe second transponder 26 and an additional FALSE timing pulse 120,which was transmitted from the first initiator 12 about three time unitspreviously. Since the A input is in a TRUE state and the TRUE timingreference pulse 118 reached the second transponder 26, which representsthe A input, the second transponder 26 will transmit a first TRUE outputpulse 124 (not shown) in about one time unit.

FIG. 13C, which represents a snapshot at twelve time units elapsed time,shows the TRUE timing reference pulse 118 reaching the third transponder28, the first TRUE output pulse 124, which was transmitted from thesecond transponder 26 about two time units previously, and an additionalTRUE timing reference pulse 118, which was transmitted from the secondinitiator 22 about two time units previously. Since the B input is in aTRUE state and the TRUE timing reference pulse 118 reached the thirdtransponder 28, which represents the B input, the third transponder 28will transmit a second TRUE output pulse 126 (not shown) in about onetime unit.

FIG. 13D, which represents a snapshot at eighteen time units elapsedtime, shows the first and the second TRUE output pulses 124, 126 and theTRUE timing reference pulse 118 reaching the first transponder 24simultaneously, and an additional FALSE timing reference pulse 120,which was transmitted from the first initiator 12 about two time unitspreviously. Since the first transponder 24 received two or more TRUEpulses simultaneously, the first transponder 24 will respond bytransmitting a third TRUE output pulse 128 (not shown) in about one timeunit.

FIG. 13E, which represents a snapshot at twenty-four time units elapsedtime, shows the third TRUE output pulse 128 an additional TRUE timingreference pulse 118 reaching the fifth transponder 122 simultaneously.Since the fifth transponder 122 received two TRUE pulses simultaneously,the fifth transponder 122 will respond by transmitting a FALSE outputpulse (not shown) in about one time unit. Since the inverting output ofthe NOR gate should be FALSE and since the fifth transponder 122provides the inverting output, the response from the fifth transponder122 is correct.

FIGS. 14A through 14F illustrate behavior of the Boolean NOR gate usingthe polychronous wave propagation system 20 with both A and B inputs ofthe NOR gate being in a FALSE state; therefore, the inverting output ofthe NOR gate should be TRUE. The states of the A and the B inputs mayhave been driven into these states by other initiators (not shown), bycircuitry driven by the aggregated initiator control information INCIfrom the control system 64, or both. FIGS. 14A, 14B, 14C, 14D, 14E, and14F may represent snapshots in time of the first wave propagation medium10 at two time units elapsed time, six time units elapsed time, eleventime units elapsed time, nineteen time units elapsed time, twenty-fourtime units elapsed time, and thirty-one nanoseconds elapsed time,respectively.

FIG. 14A, which represents a snapshot at two time units elapsed time,shows a TRUE timing reference pulse 118, which was transmitted from thesecond initiator 22 about two time units previously and a FALSE timingreference pulse 120, which was transmitted from the first initiator 12about six time units previously; therefore, the TRUE timing referencepulses 118 lag respective FALSE timing reference pulses 120 by aboutfour time units. FIG. 14B, which represents a snapshot at six time unitselapsed time, shows the FALSE timing reference pulse 120 reaching thesecond transponder 26. Since the A input is in a FALSE state and theFALSE timing reference pulse 120 reached the second transponder 26,which represents the A input, the second transponder 26 will transmit afirst FALSE output pulse 130 (not shown) in about one time unit.

FIG. 14C, which represents a snapshot at eleven time units elapsed time,shows the FALSE timing reference pulse 120 reaching the thirdtransponder 28, the first FALSE output pulse 130, which was transmittedfrom the second transponder 26 about four time units previously, and anadditional FALSE timing reference pulse 120, which was transmitted fromthe first initiator 12 about five time units previously. Since the Binput is in a FALSE state and the FALSE timing reference pulse 120reached the third transponder 28, which represents the B input, thethird transponder 28 will transmit a second FALSE output pulse 132 (notshown) in about one time unit.

FIG. 14D, which represents a snapshot at nineteen time units elapsedtime, shows the first and the second FALSE output pulses 130, 132reaching the fourth transponder 30 simultaneously, and an additionalFALSE timing reference pulse 120, which was transmitted from the firstinitiator 12 about three time units previously. Since the fourthtransponder 30 received two or more FALSE pulses simultaneously, thefourth transponder 30 will respond by transmitting a third FALSE outputpulse 134 (not shown) in about one time unit.

FIG. 14E, which represents a snapshot at twenty-four time units elapsedtime, shows the third FALSE output pulse 134 and the FALSE timingreference pulse 120 reaching the first transponder 24 simultaneously.Since the first transponder 24 received two or more FALSE pulsessimultaneously, the first transponder 24 will respond by transmitting afourth FALSE output pulse 136 (not shown) in about one time unit.

FIG. 14F, which represents a snapshot at thirty-one time units elapsedtime, shows the fourth FALSE output pulse 136 and an additional FALSEtiming reference pulse 120 reaching the fifth transponder 122simultaneously. Since the fifth transponder 122 received two FALSEpulses simultaneously, the fifth transponder 122 will respond bytransmitting a TRUE output pulse (not shown) in about one time unit.Since the inverting output of the NOR gate should be TRUE and since thefifth transponder 122 provides the inverting output, the response fromthe fifth transponder 122 is correct.

FIGS. 15A through 15E illustrate behavior of the Boolean NOR gate usingthe polychronous wave propagation system 20 with the A input of the NORgate being in a TRUE state and the B input of the NOR gate being in aFALSE state; therefore, the inverting output of the NOR gate should beFALSE. The states of the A and the B inputs may have been driven intothese states by other initiators (not shown), by circuitry driven by theaggregated initiator control information INCI from the control system64, or both. FIGS. 15A, 15B, 15C, 15D, and 15E may represent snapshotsin time of the first wave propagation medium 10 at two time unitselapsed time, nine time units elapsed time, twelve time units elapsedtime, eighteen time units elapsed time, and twenty-four time unitselapsed time, respectively.

FIG. 15A, which represents a snapshot at two time units elapsed time,shows a TRUE timing reference pulse 118, which was transmitted from thesecond initiator 22 about two time units previously and a FALSE timingreference pulse 120, which was transmitted from the first initiator 12about six time units previously; therefore, the TRUE timing referencepulses 118 lag respective FALSE timing reference pulses 120 by aboutfour time units. FIG. 15B, which represents a snapshot at nine timeunits elapsed time, shows the TRUE timing reference pulse 118 reachingthe second transponder 26 and an additional FALSE timing reference pulse120, which was transmitted from the first initiator 12 about three timeunits previously. Since the A input is in a TRUE state and the TRUEtiming reference pulse 118 reached the second transponder 26, whichrepresents the A input, the second transponder 26 will transmit thefirst TRUE output pulse 124 (not shown) in about one time unit.

FIG. 15C, which represents a snapshot at twelve time units elapsed time,shows a FALSE timing reference pulse 120 about one time unit beyond thethird transponder 28, the first TRUE output pulse 124, which wastransmitted from the second transponder 26 about two time unitspreviously, and an additional TRUE timing reference pulse 118, which wastransmitted from the second initiator 22 about two time unitspreviously. Since the B input is in a FALSE state and the FALSE timingreference pulse 120 reached the third transponder 28, which representsthe B input, the third transponder 28 is about to transmit the secondFALSE output pulse 132 (not shown).

FIG. 15D, which represents a snapshot at eighteen time units elapsedtime, shows the first TRUE output pulse 124 and the TRUE timingreference pulse 118 reaching the first transponder 24 simultaneously,and an additional FALSE timing reference pulse 120, which wastransmitted from the first initiator 12 about two time units previously.Since the first transponder 24 received two or more TRUE pulsessimultaneously, the first transponder 24 will respond by transmittingthe third TRUE output pulse 128 (not shown) in about one time unit.

FIG. 15E, which represents a snapshot at twenty-four time units elapsedtime, shows the third TRUE output pulse 128 and an additional TRUEtiming reference pulse 118 reaching the fifth transponder 122simultaneously. Since the fifth transponder 122 received two TRUE pulsessimultaneously, the fifth transponder 122 will respond by transmitting aFALSE output pulse (not shown) in about one time unit. Since theinverting output of the NOR gate should be FALSE and since the fifthtransponder 122 provides the inverting output, the response from thefifth transponder 122 is correct.

FIGS. 16A through 16E illustrate behavior of the Boolean NOR gate usingthe polychronous wave propagation system 20 with the A input of the NORgate being in a FALSE state and the B input of the NOR gate being in aTRUE state; therefore, the inverting output of the NOR gate should beFALSE. The states of the A and the B inputs may have been driven intothese states by other initiators (not shown), by circuitry driven by theaggregated initiator control information INCI from the control system64, or both. FIGS. 16A, 16B, 16C, 16D, and 16E may represent snapshotsin time of the first wave propagation medium 10 at two time unitselapsed time, six time units elapsed time, twelve time units elapsedtime, eighteen time units elapsed time, and twenty-four time unitselapsed time, respectively.

FIG. 16A, which represents a snapshot at two time units elapsed time,shows a TRUE timing reference pulse 118, which was transmitted from thesecond initiator 22 about two time units previously and a FALSE timingreference pulse 120, which was transmitted from the first initiator 12about six time units previously; therefore, the TRUE timing referencepulses 118 lag respective FALSE timing reference pulses 120 by aboutfour time units. FIG. 16B, which represents a snapshot at six time unitselapsed time, shows the FALSE timing reference pulse 120 reaching thesecond transponder 26. Since the A input is in a FALSE state and theFALSE timing reference pulse 120 reached the second transponder 26,which represents the A input, the second transponder 26 will transmitthe first FALSE output pulse 130 (not shown) in about one time unit.

FIG. 16C, which represents a snapshot at twelve time units elapsed time,shows the TRUE timing reference pulse 118 reaching the third transponder28, the first FALSE output pulse 130, which was transmitted from thesecond transponder 26 about five time units previously, and anadditional TRUE timing reference pulse 118, which was transmitted fromthe second initiator 22 about two time units previously. Since the Binput is in a TRUE state and the TRUE timing reference pulse 118 reachedthe third transponder 28, which represents the B input, the thirdtransponder 28 will transmit the second TRUE output pulse 126 (notshown) in about onetime unit.

FIG. 16D, which represents a snapshot at eighteen time units elapsedtime, shows the second TRUE output pulse 126 and a TRUE timing referencepulse 118 reaching the first transponder 24 simultaneously, and anadditional FALSE timing reference pulse 120, which was transmitted fromthe first initiator 12 about six time units previously. Since the firsttransponder 24 received two or more TRUE pulses simultaneously, thefirst transponder 24 will respond by transmitting the third TRUE outputpulse 128 (not shown) in about one time unit.

FIG. 16E, which represents a snapshot at twenty-four time units elapsedtime, shows the third TRUE output pulse 128 and an additional TRUEtiming reference pulse 118 reaching the fifth transponder 122simultaneously. Since the fifth transponder 122 received two TRUE pulsessimultaneously, the fifth transponder 122 will respond by transmitting aFALSE output pulse (not shown) in about one time unit. Since theinverting output of the NOR gate should be FALSE and since the fifthtransponder 122 provides the inverting output, the response from thefifth transponder 122 is correct.

FIGS. 17A through 20B illustrate behaviors of the polychronous wavepropagation system 20 in which asynchronous waves that have timingrelationships to one another are transmitted and propagate through thefirst wave propagation medium 10. The polychronous wave propagationsystem 20 includes the first transponder 24, the second transponder 26,the third transponder 28, and in some embodiments the fourth transponder30. The first, the second, the third, and the fourth transponders 24,26, 28, 30 may be carefully located with respect to one another toproduce desired behaviors in the polychronous wave propagation system20. Each of the first, the second, the third, and the fourthtransponders 24, 26, 28, 30 is not responsive to single pulses, but isresponsive to simultaneous pulses having the same state by transmittinga pulse with the same state. For example if the first transponder 24receives two TRUE pulses simultaneously, the first transponder 24responds by transmitting a TRUE pulse. Similarly, if the firsttransponder 24 receives two FALSE pulses simultaneously, the firsttransponder 24 responds by transmitting a FALSE pulse.

FIGS. 17A through 17E illustrate behavior of a four transponderreverberating memory cell in a TRUE state using the polychronous wavepropagation system 20 in one embodiment of the polychronous wavepropagation system 20. Typically, the polychronous wave propagationsystem 20 would provide multiple reverberating memory cells. The fourtransponder reverberating memory cell operates by programming andpreserving a state into the memory cell. The second and the fourthtransponders 26, 30 may have been previously driven into the TRUE stateby other initiators (not shown), by circuitry driven by the aggregatedinitiator control information INCI from the control system 64, or both.FIGS. 17A, 17B, 17C, 17D, and 17E may represent snapshots in time of thefirst wave propagation medium 10 at zero time units elapsed time, threetime units elapsed time, nine time units elapsed time, thirteen timeunits elapsed time, and nineteen time units elapsed time, respectively.

FIG. 17A, which represents a snapshot at zero time units elapsed time,shows the first, the second, the third, and the fourth transponders 24,26, 28, 30. The second and the fourth transponders 26, 30 are about totransmit first and second TRUE memory pulses 138, 140 (not shown),respectively. FIG. 17B, which represents a snapshot at three time unitselapsed time, shows the first and the second TRUE memory pulses 138, 140provided by the second and the fourth transponders 26, 30, respectively.FIG. 17C, which represents a snapshot at nine time units elapsed time,shows the first and the second TRUE memory pulses 138, 140simultaneously reaching the third and the first transponders 28, 24.Since the third and the first transponders 28, 24 each received two TRUEpulses simultaneously, the third and the first transponders 28, 24 willeach respond by transmitting a third TRUE memory pulse 142 (not shown)and a fourth TRUE memory pulse 144 (not shown), respectively, in aboutone time unit.

FIG. 17D, which represents a snapshot at thirteen time units elapsedtime, shows the third and the fourth TRUE memory pulses 142, 144provided by the third and the first transponders 28, 24, respectively.FIG. 17E, which represents a snapshot at nineteen time units elapsedtime, shows the third and the fourth TRUE memory pulses 142, 144simultaneously reaching the second and the fourth transponders 26, 30.Since the second and the fourth transponders 26, 30 each received twoTRUE pulses simultaneously, the second and the fourth transponders 26,30 will each respond by re-transmitting the first and the second TRUEmemory pulses 138, 140, respectively, (not shown) in about onetime unit,thereby sustaining the TRUE state, which is reverberated between thefirst, the second, the third, and the fourth transponders 24, 26, 28,30.

FIGS. 18A through 18D illustrate behavior of a four transponderreverberating memory cell in a FALSE state using the polychronous wavepropagation system 20 in one embodiment of the polychronous wavepropagation system 20. FIGS. 18A through 18D are similar to FIGS. 17Bthrough 17E, respectively, except the four transponder reverberatingmemory cell is in a FALSE state instead of a TRUE state and the first,the second, the third, and the fourth TRUE memory pulses 138, 140, 142,144 are replaced with first, second, third, and fourth FALSE memorypulses 146, 148, 150, 152, respectively.

FIG. 19 illustrates behavior of a reverberating memory doublet cellusing the polychronous wave propagation system 20 in an alternateembodiment of the polychronous wave propagation system 20. Typically,the polychronous wave propagation system 20 would provide multiplereverberating memory doublet cells. The reverberating memory doubletcell is similar to the four transponder reverberating memory cellillustrated in FIGS. 17A through 17E, except instead of reverberating asingle pair of memory pulses, such as the first and the second TRUEmemory pulses 138, 140 or the third and the fourth TRUE memory pulses142, 144, the reverberating memory doublet cell reverberates a doubletpair of memory pulses. For example, FIG. 19 may be compared to FIG. 17E,except the third and the fourth TRUE memory pulses 144, 146 are replacedwith a first doublet outer ring 154, a first doublet inner ring 156, asecond doublet outer ring 158, and a second doublet inner ring 160. Thefirst doublet outer ring 154 and the first doublet inner ring 156 areseparated by an inter-pulse interval 162. Similarly, the second doubletouter ring 158 and the second doublet inner ring 160 are separated bythe inter-pulse interval 162. The inter-pulse interval 162 isestablished when the reverberating memory doublet cell is initialized,and since the inter-pulse interval 162 is an analog variable, which istime, the inter-pulse interval 162 may represent any real number.

The reverberating memory doublet cell illustrates a powerful benefit ofthe present invention. The four transponder reverberating memory cellstores a single memory state, which is either TRUE or FALSE. However, byleveraging upon the asynchronous time delays inherent in the fourtransponder reverberating memory cell by adding the inter-pulse interval162, much information may be added.

FIGS. 20A and 20B illustrate behavior of a three transponderreverberating memory cell using the polychronous wave propagation system20 in an additional embodiment of the polychronous wave propagationsystem 20. The three transponder reverberating memory cell operates byprogramming and preserving a state into the memory cell using the first,the second and the third transponders 24, 26, 28, which may have beenpreviously driven into the TRUE states by other initiators (not shown),by circuitry driven by the aggregated initiator control information INCIfrom the control system 64, or both. FIGS. 20A and 20B may representsnapshots in time of the first wave propagation medium 10 at three timeunits elapsed time and nine time units elapsed time, respectively.

FIG. 20A, which represents a snapshot at three time units elapsed time,shows the first, the second, and the third TRUE memory pulses 138, 140,142, which were transmitted by the third, the second, and the firsttransponders 28, 26, 24, respectively. FIG. 20B, which represents asnapshot at nine time units elapsed time, shows the first and the secondTRUE memory pulses 138, 140 simultaneously reaching the firsttransponder 24, the first and the third TRUE memory pulses 138, 142simultaneously reaching the second transponder 26, and the second andthe third TRUE memory pulses 140, 142 simultaneously reaching the thirdtransponder 28. Since the third, the second, and the first transponders28, 26, 24 each received two TRUE pulses simultaneously, the third, thesecond, and the first transponders 28, 26, 24 will each respond byre-transmitting the first, the second, and the third TRUE memory pulses138, 140, 142, respectively, in about one time unit, thereby sustainingthe TRUE state, which is reverberated between the first, the second, andthe third transponders 24, 26, 28. The FALSE state may be initializedand sustained in the three transponder reverberating memory cell in asimilar manner.

FIG. 21 illustrates frequency detection behavior of the polychronouswave propagation system 20 in another embodiment of the polychronouswave propagation system 20, which includes the first and the secondtransponders 24, 26, and first transponder array 104, in whichasynchronous waves that have timing relationships to one another aretransmitted and propagate through the first wave propagation medium 10.

The first and the second transponders 24, 26 have multiple parabolas ofintersection 164 that are each representative of timing between wavepulses (not shown) transmitted from each of the first and the secondtransponders 24, 26. In other words, when either of the first and thesecond transponders 24, 26 transmits a wave pulse and the other of thefirst and the second transponders 24, 26 transmits a wave pulse after aspecific delay, a parabola of intersection 164 is representative of thespecific delay. Different values of the specific delay have differentcorresponding parabolas of intersection 164. By arranging the firsttransponder array 104 such that each transponder in the firsttransponder array 104 is located on one of the parabolas of intersection164, the first transponder array 104 may detect the value of thespecific delay, which may provide useful information. In one embodimentof the present invention, the first transponder 24 transmits a timingreference signal; therefore, successive values of the specific delay areindicative of the frequency of wave pulses transmitted by the secondtransponder 26.

FIG. 22 shows the first controllable oscillator block 96, which formspart of the first transmit circuitry 92 illustrated in FIG. 8. Oneembodiment of the first controllable oscillator block 96 is presented.The first controllable oscillator block 96 may have multiple operatingfrequency regions, such that some of the regions tend to lock onto afrequency, while other regions do not have any locking tendencies. Astimulation using at least one stimulation signal may alter the regionsand determine a stable final frequency after a brief processinginterval.

The first controllable oscillator block 96 may have at least one basinof attraction that corresponds with a stable region of the stimulationsignal. Additionally, the first controllable oscillator block 96 mayfunction outside of a basin of attraction that corresponds with anon-stable region of the stimulation signal. A stimulation having astimulation signal within a stable region will produce an output signalwithin a frequency locking range, whereas, a stimulation having astimulation signal within a non-stable region will produce an outputsignal within a non-frequency locking range.

Each stimulation may be initiated using a stimulation signal, which mayinclude the first initialization control signal ICS₁, the first forcingsignal FS₁, or both. Initialization control signals are normally appliedonly at the beginning of a stimulation, whereas forcing signals arenormally applied throughout a stimulation. A forcing signal mayestablish the basins of attraction, whereas the initiation controlsignals may determine if the stimulation falls within any of the basinsof attraction. Output frequencies are provided by one or more outputsignals after stabilization in response to a stimulation. Each outputsignal has at least one output frequency, which is processed by othercircuitry to establish relationships between different values of thestimulation signal, and their corresponding frequencies provided by theoutput signal. These relationships may be used to mimic human brainbehaviors, such as associative memory or pattern recognition.

The first initialization control signal ICS₁ may set initial conditionsof certain parameters associated with an oscillator block, such as acapacitor voltage. The first forcing signal FS₁ may include one or moreparametric forcing signals that modify behavior of circuitry within theoscillator block. The oscillator block may have multiple basins ofattraction that may be manifested as a phase-locked loop having centerfrequencies and frequency locking ranges that may be modified byspecific aspects of the initialization control signal. Afterstabilization, an output signal may lock to a particular frequency ormay have a frequency outside of a lock range, or basin of attraction. Inone embodiment of the present invention, the first controllableoscillator block 96 includes a voltage controlled oscillator neuron(VCON) oscillator coupled to a feedback filter. The VCON oscillator isan oscillator that may mimic neuron-like behaviors. The firstcontrollable oscillator block 96 may include phase-locked loopcircuitry.

FIG. 22 shows the first controllable oscillator block 96, which receivesthe first initialization control signal ICS₁ and the first forcingsignal FS₁. The first initialization control signal ICS₁ may set initialconditions of certain parameters associated with the first controllableoscillator block 96, such as one or more initial capacitor voltages. Thefirst forcing signal FS₁ may include one or more parametric forcingsignals that modify behavior of circuitry within the first controllableoscillator block 96. The first controllable oscillator block 96 providesthe first output signal OS₁, which has at least one output frequency.After stabilization, the output frequency may be locked to a specificfrequency in a basin of attraction, and may be based on the firstinitialization control signal ICS₁, the first forcing signal FS₁, orboth.

FIG. 23 shows details of the first controllable oscillator block 96illustrated in FIG. 22, according to one embodiment of the firstcontrollable oscillator block 96. A VCON oscillator 166 provides thefirst output signal OS₁ and a frequency control signal FCS to a feedbackfilter 168. The first output signal OS₁ has a frequency based on afrequency of the VCON oscillator 166, which is based on the firstinitialization control signal ICS₁, the first forcing signal FS₁, orboth. The frequency control signal FCS is based on the frequency of theVCON oscillator 166. The feedback filter 168 filters the frequencycontrol signal FCS to provide a filtered frequency control signal FFCSto the VCON oscillator 166. The feedback filter 168 may include a lowpass filter, a band pass filter, or other type of filter. The feedbackfilter 168 may have a feedback filter transfer function H(s), which is afunction of frequency and may be equal to a magnitude and phase of thefiltered frequency control signal FFCS divided by a magnitude and phaseof the frequency control signal FCS.

FIG. 24 shows details of the VCON oscillator 166 illustrated in FIG. 23.The VCON oscillator 166 mimics certain behaviors in a brain by combininga homeostatic mechanism 170, an escapement mechanism 172, and an energystorage element 174. The homeostatic mechanism 170 is a mechanism thatcan achieve or maintain equilibrium, or stability, by means ofadjustments. A voltage controlled oscillator (VCO) is an example of ahomeostatic mechanism 170. A VCO can maintain a stable output frequencyby adjusting its input voltage. The escapement mechanism 172 is amechanism that can be used to provide feedback to regulate a system. Aclassic mechanical example is an escapement gear in a pendulum clock.The escapement gear controls extraction of energy from a mainspring,weights, etc. to the pendulum at the proper point in the swing of thependulum, thereby maintaining oscillation at the proper frequency. Anelectrical circuit having a positive resistance region and a negativeresistance region can serve as an escapement mechanism by providingbasins of attraction formed by coupling the escapement mechanism 172with other circuitry, such as the homeostatic mechanism 170 and theenergy storage element 174. The escapement mechanism 172 may have atleast one positive resistance region, at least one negative resistanceregion, or both. A tunnel diode is an example of an escapement mechanismhaving a positive resistance region and a negative resistance region. Aresistor is an example of an escapement mechanism having only a positiveresistance region.

Fundamentals of using a polychronous wave propagation system 20 toperform arithmetic computations will be described using the two-nodepolychronous wave propagation system 20 illustrated in FIG. 21. Aspreviously mentioned, FIG. 21 illustrates the polychronous wavepropagation system 20 having the first and the second transponders 24,26 and the first transponder array 104, in which asynchronous waves thathave timing relationships to one another are transmitted and propagatethrough the first wave propagation medium 10 with a propagation velocity(PV). The first and the second transponders 24, 26 have multipleparabolas of intersection 164 that are each representative of timingbetween wave pulses (not shown) transmitted from each of the first andthe second transponders 24, 26. In the example shown, the firsttransponder array 104 includes a corresponding transponder for each ofthe possible parabolas of intersection 164 that fall between the firstand the second transponders 24, 26. Therefore, asynchronous waves havingtiming relationships to one another and are transmitted and propagatedthrough the first wave propagation medium 10 associated with the firsttransponder array 104 are detectable. Each of the transponders in thefirst transponder array 104 may be representative of a result of anarithmetical computation.

In general, the polychronous wave propagation system 20 may providepolychronous activity that may be analogous to polychronous activity ina brain. The polychronous wave propagation system 20 may includemultiple initiators, such as those provided by the first and the secondtransponders 24, 26, disposed in the first wave propagation medium 10,and may include multiple responders, such as those provided by the firstand the second transponders 24, 26, disposed in the first wavepropagation medium 10. Each of the initiators may transmit waves havingencoded information through the first wave propagation medium 10. Atleast a first parabola of intersection 164 may be defined in the firstwave propagation medium 10 based on timing of transmission of theencoded information from each of a first pair of the initiators. Theparabolas of intersection 164 may be representative of an arithmeticalrelationship between the transmission of the encoded information fromthe pair of the initiators. A corresponding one of the responders may belocated on each of the parabolas of intersection 164. As such, eachcorresponding one of the responders may be capable of detectinginformation based on timing between reception of encoded informationfrom each of the first pair of initiators. Operational behavior of thepolychronous wave propagation system 20 may be based on the detectedinformation. The operational behavior may include arithmeticalcomputations, such as additions, subtractions, multiplications,divisions, modular arithmetic, the like, or any combination thereof.

The detected information associated with the corresponding one of theresponders may be based on simultaneous reception of the encodedinformation from the first pair of initiators. As previously mentioned,the first wave propagation medium 10 may include a solid material, alongest dimension of the first wave propagation medium 10 may be lessthan about one meter, the first wave propagation medium 10 may form anano-structure, the multiple initiators may be disposed in fixedpositions in the first wave propagation medium 10, the multipleresponders may be disposed in fixed positions in the first wavepropagation medium 10, or any combination thereof.

FIG. 25 shows a two-dimensional Cartesian coordinate system overlaidover the polychronous wave propagation system 20 illustrated in FIG. 21,such that the first transponder 24 is located at the origin and thesecond transponder 26 is located on the dependent (vertical) axis. Theindependent (horizontal) axis is representative of elapsed time t ofwavefronts of propagated waves from the first and the secondtransponders 24, 26. The vertical axis is representative of a spatialvariable which is representative of the locations of the wavefronts ofthe propagated waves on the line segment between the first and thesecond transponders 24, 26. The first and the second transponders 24, 26are separated by a distance D from one another. Therefore, thecoordinates of the first transponder 24 are (0, 0) and the coordinatesof the second transponder 26 are 0, D). As such, the locations of thewavefronts of the propagated waves relative to time may be representedby linear equations, as illustrated in FIG. 25 and subsequent figures.An intersection of the wavefronts of the propagated waves on the linesegment between the first and the second transponders 24, 26 may bedetected by one of the transponders in the first transponder array 104(FIG. 21). This intersection occurs at an intersection spatial valueξ_(INT).

FIG. 25 shows first and second spatial variables ξ₁, ξ₂ of wavesinitiated from the first and the second transponders 24, 26 (FIG. 21),respectively, at t=0 according to one embodiment of the polychronouswave propagation system 20 illustrated in FIG. 21. EQ. 1 and EQ. 2 areequations representing the first and the second spatial variablesrespectively, as shown below.

ξ₁ =t(PV).  EQ. 1:

ξ₂ =D−t(PV).  EQ. 2:

For the purposes of illustration, the PV is assumed to be one unit ofdistance per one unit of time for the following illustrations. As such,EQ. 1 and EQ. 2 reduce to EQ. 1A and EQ. 2A, respectively.

ξ₁ =t.  EQ. 1A

ξ₂ =D−t.  EQ. 2A:

The first and the second spatial variables ξ₁, ξ₂ intersect at a firstintersection point 176, such that ξ_(INT)=ξ₁=ξ₂. Setting EQ. 1A equal toEQ. 2A gives D−t=t, therefore, t=D/2. Substituting back into EQ. 1Agives =ξ₁=t=D/2=ξ_(INT). Since D/2 is midway between the first and thesecond transponders 24, 26, the intersection spatial value ξ_(INT) isequal to a middle spatial value ξ_(MV).

FIG. 26 shows third and fourth spatial variables ξ₃, ξ₄ of wavesinitiated from the first and the second transponders 24, 26 (FIG. 21),respectively, at t=x according to an alternate embodiment of thepolychronous wave propagation system 20 illustrated in FIG. 21. EQ. 3and EQ. 4 are equations representing the first and the second spatialvariables ξ₃, ξ₄, respectively, as shown below.

ξ₃ =t−x (for t>x).  EQ. 3:

ξ₄ =D+x−t (for t>x).  EQ. 4:

The third and the fourth spatial variables ξ₃, ξ₄ intersect at a secondintersection point 178, such that ξ_(INT)=ξ₃=ξ₄. Setting EQ. 3 equal toEQ. 4 gives D+x−t=t−x, therefore, t=x+D/2. Substituting back into EQ. 3gives =ξ₃=t−x=x+D/2−x=D/2=ξ_(INT). Since D/2 is midway between the firstand the second transponders 24, 26, the intersection spatial valueξ_(INT) is equal to the middle spatial value ξ_(MV).

FIG. 27 shows fifth and sixth spatial variables ξ₅, ξ₆ of wavesinitiated from the first and the second transponders 24, 26 (FIG. 21),respectively, at t=y according to an alternate embodiment of thepolychronous wave propagation system 20 illustrated in FIG. 21. EQ. 5and EQ. 6 are equations representing the fifth and the sixth spatialvariables ξ₅, ξ₆, respectively, as shown below.

ξ₅ =t−y (for t>y).  EQ. 5:

ξ₆ =D+y−t (for t>y).  EQ. 6:

The fifth and the sixth spatial variables ξ₅, ξ₆ intersect at a thirdintersection point 178, such that ξ_(INT)=ξ₅=ξ₆. Setting EQ. 5 equal toEQ. 6 gives D+y−t=t−y, therefore, t=y+D/2. Substituting back into EQ. 5gives =4=t−y=y+D/2−y=D/2=ξ_(INT). Since D/2 is midway between the firstand the second transponders 24, 26, the intersection spatial valueξ_(INT) is equal to the middle spatial value ξ_(MV).

FIG. 28 shows a combination of the first, the second, the third, thefourth, the fifth and the sixth spatial variables ξ₁, ξ₂, ξ₃, ξ₄, ξ₅,ξ₆, illustrated in FIGS. 25-27 according to one embodiment of thepolychronous wave propagation system 20 illustrated in FIG. 21. As such,waves are initiated from the first and the second transponders 24, 26(FIG. 21), respectively, at t=0, at t=x, and at t=y. Intersections ofthese waves occur at nine intersection points, as illustrated.Specifically, the first and the second spatial variables intersect atthe first intersection point 176, such that ξ_(INT)=ξ₁=ξ₂=ξ_(MV). Thethird and the fourth spatial variables ξ₃, ξ₄ intersect at the secondintersection point 178, such that ξ_(INT)=ξ₃=ξ₄=ξ_(MV). The fifth andthe sixth spatial variables ξ₅, ξ₆ intersect at the third intersectionpoint 180, such that ξ_(INT)=ξ₅=ξ₆=ξ_(MV).

The second and the third spatial variables ξ₂, ξ₃ intersect at a fourthintersection point 182, such that ξ_(INT)=ξ₂=ξ₃. The intersectionspatial value ξ_(INT) is equal to a first spatial value ξ_(V1). SettingEQ. 2A equal to EQ. 3 gives D−t=t−x, therefore, t=x/2+D/2. Substitutingback into EQ. 2A gives =ξ₂=D−t=D−x/2−D/2=D/2−x/2=ξ_(V1). Therefore,

ξ_(V1) =D/2−x/2.  EQ. 7:

The second and the fifth spatial variables ξ₂, ξ₅ intersect at a fifthintersection point 184, such that ξ_(INT)=ξ₂=₅. The intersection spatialvalue ξ_(INT) is equal to a second spatial value ξ_(V2). Setting EQ. 2Aequal to EQ. 5 gives D−t=t−y, therefore, t=y/2+D/2. Substituting backinto EQ. 2A gives =ξ₂=D−t=D−y/2−D/2=D/2−y/2=ξ_(V2).

The fourth and the fifth spatial variables ξ₄, ξ₅ intersect at a sixthintersection point 186, such that ξ_(INT)=ξ₄=ξ₅. The intersectionspatial value ξ_(INT) is equal to a third spatial value ξ_(V3). SettingEQ. 4 equal to EQ. 5 gives D+x−t=t−y, therefore, t=x/2+y/2+D/2.Substituting back into EQ. 5 gives=ξ₅=t−y=D/2+x/2+y/2−y=D/2+(x−y)/2=ξ_(V3). Therefore,

ξ_(V3) =D/2+(x−y)/2.  EQ. 8:

The first and the sixth spatial variables ξ₁, ξ₆ intersect at a seventhintersection point 188, such that ξ_(INT)=ξ₁=ξ₆. The intersectionspatial value ξ_(INT) is equal to a fourth spatial value ξ_(V4). SettingEQ. 1A equal to EQ. 6 gives t=D+y−t, therefore, t=y/2+D/2. Substitutingback into EQ. 1A gives =ξ₁=t=D/2+y/2=ξ₄.

The third and the sixth spatial variables ξ₃, ξ₆ intersect at an eighthintersection point 190, such that ξ_(INT)=ξ₃=ξ₆. The intersectionspatial value ξ_(INT) is equal to a fifth spatial value ξ_(V5). SettingEQ. 3 equal to EQ. 6 gives t−x=D+y−t, therefore, t=x/2+y/2+D/2.Substituting back into EQ. 3 gives=ξ₃=t−x=D/2+x/2+y/2−x=D/2+(y−x)/2=ξ_(V5). Therefore,

ξ_(V5) =D/2+(y−x)/2.  EQ. 9:

The first and the fourth spatial variables intersect at a ninthintersection point 192, such that ξ_(INT)=ξ₁=ξ₄. The intersectionspatial value ξ_(INT) is equal to a sixth spatial value ξ_(V6). SettingEQ. 1A equal to EQ. 4 gives t=D+x−t, therefore, t=x/2+D/2. Substitutingback into EQ. 1A gives =ξ₁=t=D/2+x/2=ξ_(V6). Therefore,

ξ_(V6) =D/2+x/2.  EQ. 10:

FIG. 29 is used to illustrate the arithmetical computation ofsubtraction. In this regard, FIG. 29 shows a combination of the third,the fourth, the fifth and the sixth spatial variables ξ₃, ξ₄, ξ₅, ξ₆illustrated in FIG. 28 according to one embodiment of the polychronouswave propagation system 20 illustrated in FIG. 21. As such, waves areinitiated from the first and the second transponders 24, 26 (FIG. 21) att=x and at t=y. Intersections of these waves occur at four intersectionpoints, as illustrated. As previously mentioned, the third and thefourth spatial variables ξ₃, ξ₄ intersect at the second intersectionpoint 178, such that ξ_(INT)=ξ₃=ξ₄=ξ_(MV). The fifth and the sixthspatial variables ξ₅, ξ₆ intersect at the third intersection point 180,such that ξ_(INT)=ξ₅=ξ₆=ξ_(MV).

The fourth and the fifth spatial variables ξ₄, ξ₅ intersect at the sixthintersection point 186, such that ξ_(INT)=ξ₄=ξ₅. The intersectionspatial value ξ_(INT) is equal to the third spatial value ξ_(V3), whichis equal to D/2+(x−y)/2 as shown in EQ. 8. Therefore, a transponder inthe first transponder array 104 (FIG. 21) located at the third spatialvalue ξ_(V3) may detect the waves that are initiated from the first andthe second transponders 24, 26 (FIG. 21) at t=x and at t=y. As such, thetransponder in the first transponder array 104 (FIG. 21) located at thethird spatial value ξ_(V3) may be representative of subtraction as adifference between x and y. Specifically, the transponder in the firsttransponder array 104 (FIG. 21) located at the third spatial valueξ_(V3) may be representative of D/2+(x−y)/2.

The third and the sixth spatial variables ξ₃, ξ₆ intersect at the eighthintersection point 190, such that ξ_(INT)=ξ₃=₆. The intersection spatialvalue ξ_(INT) is equal to the fifth spatial value ξ_(V5), which is equalto D/2+(y−x)/2 as shown in EQ. 9. Therefore, a transponder in the firsttransponder array 104 (FIG. 21) located at the fifth spatial valueξ_(V5) may detect the waves that are initiated from the first and thesecond transponders 24, 26 (FIG. 21) at t=x and at t=y. As such, thetransponder in the first transponder array 104 (FIG. 21) located at thefifth spatial value ξ_(V5) may be representative of subtraction as adifference between x and y. Specifically, the transponder in the firsttransponder array 104 (FIG. 21) located at the fifth spatial valueξ_(V5) may be representative of D/2+(y−x)/2. In general, waves thatinitiated from the first and the second transponders 24, 26 (FIG. 21) ata first time, which is representative of a first magnitude of a firstvariable, and initiated at a second time, which is representative of asecond magnitude of a second variable, may be received by at least onetransponder in the first transponder array 104 (FIG. 21) that isrepresentative of a difference between the first magnitude and thesecond magnitude. In this regard, timing of a first transmission ofencoded information from a first pair of initiators, which are providedby the first and the second transponders 24, 26, may be based on thefirst magnitude of the first variable. Timing of a second transmissionof encoded information from the first pair of initiators may be based onthe second magnitude of the second variable. At least one of theparabolas of intersection 164 may be based on the difference between thefirst magnitude and the second magnitude.

FIG. 30 is used to illustrate the arithmetical computation ofmultiplication. In this regard, FIG. 30 shows a combination of thethird, the fourth, the fifth and the sixth spatial variables ξ₃, ξ₄, ξ₅,ξ₆ illustrated in FIG. 28 according to one embodiment of thepolychronous wave propagation system 20 illustrated in FIG. 21. As such,waves are initiated from the first and the second transponders 24, 26(FIG. 21) at t=x and at t=y.

Intersections of these waves occur at four intersection points, asillustrated. As previously mentioned, the third and the fourth spatialvariables ξ₃, ξ₄ intersect at the second intersection point 178, suchthat ξ_(INT)=ξ₃=ξ₄=ξ_(MV). The fifth and the sixth spatial variablesintersect at the third intersection point 180, such thatξ_(INT)=ξ₅=ξ₆=ξ_(MV).

Multiplication may be performed by adding multiple times. Specifically,the waves that are initiated from the first and the second transponders24, 26 (FIG. 21) at t=y are based on delaying initiation of the waves“k” times x after the waves that are initiated from the first and thesecond transponders 24, 26 (FIG. 21) at t=x. For example, at t=x, aseparate timer could be utilized to track when t=x+x, when t=x+x+x, whent=x+x+x+x, and so on until t=x+kx. When t=x+kx, t=y. Therefore,

y=(k+1)x.  EQ. 11:

Substituting EQ. 11 into EQ. 8 produces EQ. 12, as shown below.

ξ_(V3) =D/2+(x−(k+1)x)/2=D/2−(kx)/2.  EQ. 12:

Similarly, substituting EQ. 11 into EQ. 9 produces EQ. 13, as shownbelow.

ξ_(V5) =D/2+((k+1)x−x)/2=D/2+(kx)/2.  EQ. 13:

The fourth and the fifth spatial variables ξ₄, ξ₅ intersect at the sixthintersection point 186, such that ξ_(INT)=ξ₄=ξ₅. The intersectionspatial value ξ_(INT) is equal to the third spatial value ξ_(V3), whichis equal to D/2−(kx)/2 as shown in EQ. 12. Therefore, a transponder inthe first transponder array 104 (FIG. 21) located at the third spatialvalue may detect the waves that are initiated from the first and thesecond transponders 24, 26 (FIG. 21) at t=x and at t=y. As such, thetransponder in the first transponder array 104 (FIG. 21) located at thethird spatial value ξ_(V3) may be representative of multiplication as aproduct between x and k. Specifically, the transponder in the firsttransponder array 104 (FIG. 21) located at the third spatial valueξ_(V3) may be representative of D/2−(kx)/2.

The third and the sixth spatial variables ξ₃, ξ₆ intersect at the eighthintersection point 190, such that ξ_(INT)=ξ₃=ξ₆. The intersectionspatial value DINT is equal to the fifth spatial value ξ_(V5), which isequal to D/2+(kx)/2 as shown in EQ. 13. Therefore, a transponder in thefirst transponder array 104 (FIG. 21) located at the fifth spatial valueξ_(V5) may detect the waves that are initiated from the first and thesecond transponders 24, 26 (FIG. 21) at t=x and at t=y. As such, thetransponder in the first transponder array 104 (FIG. 21) located at thefifth spatial value ξ_(V5) may be representative of multiplication as aproduct between x and k. Specifically, the transponder in the firsttransponder array 104 (FIG. 21) located at the fifth spatial valueξ_(V5) may be representative of D/2+(kx)/2. In general, waves thatinitiated from the first and the second transponders 24, 26 (FIG. 21) ata first time, which is representative of a first magnitude of a firstvariable, and initiated at a second time, which is representative of asecond magnitude of a second variable, may be received by at least onetransponder in the first transponder array 104 (FIG. 21) that isrepresentative of a product between the first magnitude and the secondmagnitude. In this regard, timing of a first transmission of encodedinformation from a first pair of initiators, which are provided by thefirst and the second transponders 24, 26, may be based on the firstmagnitude of the first variable. Timing of a second transmission ofencoded information from the first pair of initiators may be based onthe second magnitude of the second variable. At least one of theparabolas of intersection 164 may be based on the product between thefirst magnitude and the second magnitude.

FIG. 31 may be used to illustrate the arithmetical computation ofdivision. In this regard, FIG. 31 shows a combination of the third, thefourth, the fifth and the sixth spatial variables ξ₃, ξ₄, ξ₅, ξ₆illustrated in FIG. 28 according to one embodiment of the polychronouswave propagation system 20 illustrated in FIG. 21. As such, waves areinitiated from the first and the second transponders 24, 26 (FIG. 21) att=x and at t=y. Intersections of these waves occur at four intersectionpoints, as illustrated. As previously mentioned, the third and thefourth spatial variables ξ₃, ξ₄ intersect at the second intersectionpoint 178, such that ξ_(INT)=ξ₃=ξ₄=ξ_(MV). The fifth and the sixthspatial variables ξ₅, ξ₆ intersect at the third intersection point 180,such that ξ_(INT)=ξ₅=ξ₆=ξ_(MV).

Division may be performed by subtracting multiple times. Specifically,the waves that are initiated from the first and the second transponders24, 26 (FIG. 21) at t=y may be based on delaying initiation of suchwaves after the waves that are initiated from the first and the secondtransponders 24, 26 (FIG. 21) at t=x by an amount based on x divided bya divisor “d”. For example, at t=0, a separate timer could be utilizedto track when t=d, when t=d+d, when t=d+d+d, and so on until t=x. Eachtime t reaches a multiple of d, a count “q” is incremented. As a result,“q” effectively keeps track of the number of times d is subtracted fromx. In this regard, x=qd. When t=x+q, then t=y and waves are initiatedfrom the first and the second transponders 24, 26 (FIG. 21). Therefore,

y=x+q=x+x/d.  EQ. 14:

Substituting EQ. 14 into EQ. 8 produces EQ. 15, as shown below.

ξ_(V3) =D/2+(x−(x+x/d))/2=D/2−x/2d.  EQ. 15:

Similarly, substituting EQ. 14 into EQ. 9 produces EQ. 16, as shownbelow.

ξ_(V5) =D/2+((x+x/d)−x)/2=D/2+x/2d.  EQ. 16:

The fourth and the fifth spatial variables ξ₄, ξ₅ intersect at the sixthintersection point 186, such that ξ_(INT)=ξ₄=ξ₅. The intersectionspatial value ξ_(INT) is equal to the third spatial value ξ_(V3), whichis equal to D/2−x/2d as shown in EQ. 15. Therefore, a transponder in thefirst transponder array 104 (FIG. 21) located at the third spatial valueξ_(V3) may detect the waves that are initiated from the first and thesecond transponders 24, 26 (FIG. 21) at t=x and at t=y. As such, thetransponder in the first transponder array 104 (FIG. 21) located at thethird spatial value ξ_(V3) may be representative of division as aquotient between x and d. Specifically, the transponder in the firsttransponder array 104 (FIG. 21) located at the third spatial valueξ_(V3) may be representative of x divided by d, or D/2−x/2d.

The third and the sixth spatial variables ξ₃, ξ₆ intersect at the eighthintersection point 190, such that ξ_(INT)=ξ₃=ξ₆. The intersectionspatial value ξ_(INT) is equal to the fifth spatial value ξ_(V5), whichis equal to D/2+x/2d as shown in EQ. 16. Therefore, a transponder in thefirst transponder array 104 (FIG. 21) located at the fifth spatial valueξ_(V5) may detect the waves that are initiated from the first and thesecond transponders 24, 26 (FIG. 21) at t=x and at t=y. As such, thetransponder in the first transponder array 104 (FIG. 21) located at thefifth spatial value ξ_(V5) may be representative of division as aquotient between x and d. Specifically, the transponder in the firsttransponder array 104 (FIG. 21) located at the fifth spatial valueξ_(V5) may be representative of x divided by d, or D/2+x/2d. In general,waves that initiated from the first and the second transponders 24, 26(FIG. 21) at a first time, which is representative of a first magnitudeof a first variable, and initiated at a second time, which isrepresentative of a second magnitude of a second variable, may bereceived by at least one transponder in the first transponder array 104(FIG. 21) that is representative of a quotient between the firstvariable and the second variable. In this regard, timing of a firsttransmission of encoded information from a first pair of initiators,which are provided by the first and the second transponders 24, 26, maybe based on the first magnitude of the first variable. Timing of asecond transmission of encoded information from the first pair ofinitiators may be based on the second magnitude of the second variable.At least one of the parabolas of intersection 164 may be based on thequotient between the first magnitude and the second magnitude. Thequotient may be based on the first magnitude divided by the secondmagnitude.

FIG. 32 may be used to illustrate the arithmetical computation ofaddition. In this regard, FIG. 32 shows a graph illustrating acombination of the first and the second spatial variables ξ₁, ξ₂illustrated in FIG. 25, and illustrating seventh and eighth spatialvariables ξ₇, ξ₈ of the line segment between the first and the secondtransponders 24, 26 illustrated in FIG. 21 versus time t according to analternate embodiment of the polychronous wave propagation system 20illustrated in FIG. 21. As such, waves are initiated from the first andthe second transponders 24, 26 (FIG. 21) at t=0 and at t=(x+y).Intersections of these waves occur at four intersection points, asillustrated.

EQ. 17 and EQ. 18 are equations representing the seventh and the eighthspatial variables respectively, as shown below.

ξ₇ =t−(x+y) (for t>(x+y)).  EQ. 17:

ξ₈ =D+(x+y)−t (for t>(x+y)).  EQ. 18:

The seventh and the eighth spatial variables ξ₇, ξ₈ intersect at a tenthintersection point 194, such that ξ_(INT)=ξ₇=ξ₈. Setting EQ. 17 equal toEQ. 18 gives D+(x+y)−t=t−(x+y), therefore, t=(x+y)+D/2. Substitutingback into EQ. 17 gives =ξ₇=t−(x+y)=(x+y)+D/2−(x+y)=D/2=ξ_(INT). SinceD/2 is midway between the first and the second transponders 24, 26, theintersection spatial value ξ_(INT) is equal to the middle spatial valueξ_(MV).

The second and the seventh spatial variables ξ₂, ξ₇ intersect at aneleventh intersection point 196, such that ξ_(INT)=ξ₂=ξ₇. Theintersection spatial value ξ_(INT) is equal to a seventh spatial valueξ_(V7). Setting EQ. 2A equal to EQ. 17 gives D−t=t−(x+y), therefore,t=x/2+y/2+D/2. Substituting back into EQ. 17 gives=ξ₇=t−(x+y)=D/2+x/2+y/2−(x+y)=D/2−x/2−y/2=ξ_(V7). Therefore,

ξ_(V7) =D/2−(x+y)/2.  EQ. 19:

The first and the eighth spatial variables ξ₁, ξ₈ intersect at a twelfthintersection point 198, such that ξ_(INT)=ξ₁=ξ₈. The intersectionspatial value INT is equal to an eighth spatial value ξ_(V8). SettingEQ. 1A equal to EQ. 18 gives t=D+(x+y)−t, therefore, t=x/2+y/2+D/2.Substituting back into EQ. 1A gives ξ₁=t=x/2+y/2+D/2=ξ_(V8). Therefore,

ξ_(V8) =D/2+(x+y)/2.  EQ. 20:

The second and the seventh spatial variables ξ₂, ξ₇ intersect at theeleventh intersection point 196, such that ξ_(INT)=ξ₂=ξ₇. Theintersection spatial value ξ_(INT) is equal to the seventh spatial valueξ_(V7), which is equal to D/2−(x+y)/2 as shown in EQ. 19. Therefore, atransponder in the first transponder array 104 (FIG. 21) located at theseventh spatial value ξ_(V7) may detect the waves that are initiatedfrom the first and the second transponders 24, 26 (FIG. 21) at t=0 andat t=(x+y). As such, the transponder in the first transponder array 104(FIG. 21) located at the seventh spatial value ξ_(V7) may berepresentative of addition as a sum between x and y. Specifically, thetransponder in the first transponder array 104 (FIG. 21) located at theseventh spatial value ξ_(V7) may be representative of D/2−(x+y)/2.

The first and the eighth spatial variables ξ₁, ξ₈ intersect at thetwelfth intersection point 198, such that ξ_(INT)=ξ₁=ξ₈. Theintersection spatial value ξ_(INT) is equal to the eighth spatial valueξ_(V8), which is equal to D/2+(x+y)/2 as shown in EQ. 20. Therefore, atransponder in the first transponder array 104 (FIG. 21) located at theeighth spatial value ξ_(V8) may detect the waves that are initiated fromthe first and the second transponders 24, 26 (FIG. 21) at t=0 and att=(x+y). As such, the transponder in the first transponder array 104(FIG. 21) located at the eighth spatial value ξ_(V8) may berepresentative of addition as a sum between x and y. Specifically, thetransponder in the first transponder array 104 (FIG. 21) located at theeighth spatial value ξ_(V8) may be representative of D/2+(x+y)/2. Ingeneral, a first transmission of encoded information may be sent from afirst pair of initiators, which are provided by the first and the secondtransponders 24, 26. Timing of a second transmission of encodedinformation from the first pair of initiators may be based on a firstmagnitude of a first variable and a second magnitude of a secondvariable. At least one of the parabolas of intersection 164 may be basedon the sum between the first magnitude and the second magnitude.

Results of the arithmetic computations using the polychronous wavepropagation system 20 illustrated in FIGS. 25 though 32 may be stored bycombining the two-node polychronous wave propagation system 20illustrated in FIG. 21 with the three transponder reverberating memorycell illustrated in FIGS. 20A and 20B. FIG. 33 shows a polychronous wavepropagation system 20 that is capable of performing and storing theresults of arithmetic computations according to one embodiment of thepolychronous wave propagation system 20. As previously described, thethree transponder reverberating memory cell illustrated in FIGS. 20A and20B includes the first transponder 24, the second transponder 26, andthe third transponder 28 disposed in the first wave propagation medium10. The three transponder reverberating memory cell may storeinformation by simultaneously transmitting waves from the transponders24, 26, 28, which are located at the vertices of an equilateraltriangle. As such, the waves transmitted from any two of thetransponders 24, 26, 28 reaches the third of the transponders 24, 26, 28simultaneously. Therefore, all three of the transponders 24, 26, 28receive two waves simultaneously. If each of the transponders 24, 26, 28then responds to the simultaneous reception of the two waves bytransmitting another wave, then the transponders 24, 26, 28 continue toreverberate with a period equal to the distance D between any two of thetransponders 24, 26, 28 divided by the propagation velocity PV. If afirst set of waves, called an alpha set, is initially transmitted at t=0and another set of waves, called a beta set, is initially transmitted att=x to create a doublet, then the transponders 24, 26, 28 willreverberate with two sets of waves per period, namely the alpha set andthe beta set. The time between two adjacent alpha sets is equal to D/PV,the time between an alpha set and the next beta set is equal to x, thetime between a beta set and the next alpha set is equal to (D/PV)−x, andthe time between two adjacent beta sets is equal to D/PV. Therefore, ifPV=1, wave sets will be transmitted at t=(0, x, D, D+x, 2D, 2D+x, 3D,3D+x, etc).

The polychronous wave propagation system 20 illustrated in FIG. 33further includes the first transponder array 104 disposed in the firstwave propagation medium 10 between the first transponder 24 and thesecond transponder 26, a second transponder array 200 disposed in thefirst wave propagation medium 10 between the first transponder 24 andthe third transponder 28, and a third transponder array 202 disposed inthe first wave propagation medium 10 between the second transponder 26and the third transponder 28. As previously mentioned, each transponderin the first transponder array 104 is located on one of the parabolas ofintersection 164 (FIG. 21) associated with the first transponder 24 andthe second transponder 26. Similarly, each transponder in the secondtransponder array 200 is located on one of the parabolas of intersection164 (not shown) associated with the first transponder 24 and the thirdtransponder 28, and each transponder in the third transponder array 202is located on one of the parabolas of intersection 164 (not shown)associated with the second transponder 26 and the third transponder 28.

FIG. 34 shows two sets of waves transmitted from the transponders 24,26, 28 illustrated in FIG. 33 according to an exemplary embodiment ofthe polychronous wave propagation system. The two sets of waves arerepresentative of the subtraction computation illustrated in FIG. 29. Afirst set 204 of waves is transmitted at t=x simultaneously from thetransponders 24, 26, 28, and a second set 206 of waves is transmitted att=y simultaneously from the transponders 24, 26, 28. FIG. 34 shows thesets 204, 206 of waves arriving at selected transponders in thetransponder arrays 104, 200, 202 that represent the parabolas ofintersection 164 (not shown) associated with differences between x andy. As such, each of the selected transponders is selected bysimultaneously receiving two waves from the sets 204, 206 of waves. Dueto symmetry in the arrangement of the transponder arrays 104, 200, 202,the selected transponders form two reverberating memory cells. In thisregard, the results of the subtraction computation may be stored. Any ofthe computations illustrated in FIGS. 29-32 may be stored in a similarmanner. FIG. 35 shows a set of waves transmitted from the selectedtransponders illustrated in FIG. 33 that identify results of thesubtraction illustrated in FIG. 34. The waves will re-trigger theselected transponders to provide the reverberating memory cells.

In general, in an exemplary embodiment of the polychronous wavepropagation system 20, the polychronous wave propagation system 20 mayinclude a first group of initiators, which may be provided by thetransponders 24, 26, 28, disposed in the first wave propagation medium10. In addition, the polychronous wave propagation system 20 may furtherinclude a first group of responders, which may be provided bytransponders in the transponder arrays 104, 200, 202, disposed in thefirst wave propagation medium 10. The first group of initiators maytransmit at least two propagated waves and the first group of respondersmay receive the propagated waves. Relative timing may be associated withinterference patterns of energy between the propagated waves.Operational behavior of the polychronous wave propagation system 20 maybe based on the relative timing and distances between the first group ofinitiators and the first group of responders. Operation behavior of thepolychronous wave propagation system 20 may include one or morearithmetical computation.

Further, the polychronous wave propagation system 20 may store theresults of the arithmetical computation by providing a reverberatingmemory cell. In general, by expanding upon the exemplary embodiment ofthe polychronous wave propagation system 20 presented above, areverberating memory cell may be provided. The polychronous wavepropagation system 20 may further include a second group of initiators,which may be provided by transponders in the transponder arrays 104,200, 202, disposed in the first wave propagation medium 10. In addition,the polychronous wave propagation system 20 may further include a secondgroup of responders, which may be provided by transponders in thetransponder arrays 104, 200, 202, disposed in the first wave propagationmedium 10. The second group of initiators may simultaneously transmit aset of at least three propagated waves and the second group ofresponders may receive the set of at least three propagated waves. Uponreceipt of the set of at least three propagated waves by the secondgroup of responders, the second group of initiators may re-transmit theset of at least three propagated waves to provide the reverberatingmemory cell. The second group of initiators may continue to re-transmitthe set of at least three propagated waves as long as the reverberatingmemory cell needs to store the results of the arithmetical computation.The first group of responders may be identical to the second group ofresponders, as shown in FIG. 34 and FIG. 35.

FIG. 36 shows a three-dimensional view of the polychronous wavepropagation system 20 having a group of wave propagation media accordingto one embodiment of the polychronous wave propagation system 20. Thegroup of wave propagation media include the first wave propagationmedium 10, a second wave propagation medium 208, a third wavepropagation medium 210, and up to and including an N^(TH) wavepropagation medium 212. The wave propagation media 10, 208, 210, 212 maybe stacked on top of one another as shown or may be arranged in anymanner. The transponders 24, 26, 28 and the transponder arrays 104, 200,202 may be disposed in the first wave propagation medium 10 similarly tothe transponders 24, 26, 28 and the transponder arrays 104, 200, 202illustrated in FIG. 33.

FIG. 37 shows details of the second wave propagation medium 208illustrated in FIG. 36 according to one embodiment of the second wavepropagation medium 208. A second medium first transponder 214, a secondmedium second transponder 216, a second medium third transponder 218, asecond medium first transponder array 220, a second medium secondtransponder array 222, and a second medium third transponder array 224are disposed in the second wave propagation medium 208 as illustrated inFIG. 37 in a similar manner to the first medium transponders 24, 26, 28and the first medium transponder arrays 104, 200, 202 illustrated inFIG. 36. As such, the second medium transponders 214, 216, 218 arelocated at the vertices of an equilateral triangle. The second mediumfirst transponder array 220 is located between the second medium firsttransponder 214 and the second medium second transponder 216. The secondmedium second transponder array 222 is located between the second mediumfirst transponder 214 and the second medium third transponder 218. Thesecond medium third transponder array 224 is located between the secondmedium second transponder 216 and the second medium third transponder218. In this regard, the second medium transponders 214, 216, 218 andthe second medium transponder arrays 220, 222, 224 may behave in asimilar manner to the first medium transponders 24, 26, 28 and the firstmedium transponder arrays 104, 200, 202 illustrated in FIG. 36.

FIG. 38 shows details of the third wave propagation medium 210illustrated in FIG. 36 according to one embodiment of the third wavepropagation medium 210. A third medium first transponder 226, a thirdmedium second transponder 228, a third medium third transponder 230, athird medium first transponder array 232, a third medium secondtransponder array 234, and a third medium third transponder array 236are disposed in the third wave propagation medium 210 as illustrated inFIG. 38 in a similar manner to the first medium transponders 24, 26, 28and the first medium transponder arrays 104, 200, 202 illustrated inFIG. 36. As such, the third medium transponders 226, 228, 230 arelocated at the vertices of an equilateral triangle. The third mediumfirst transponder array 232 is located between the third medium firsttransponder 226 and the third medium second transponder 228. The thirdmedium second transponder array 234 is located between the third mediumfirst transponder 226 and the third medium third transponder 230. Thethird medium third transponder array 236 is located between the thirdmedium second transponder 228 and the third medium third transponder230. In this regard, the third medium transponders 226, 228, 230 and thethird medium transponder arrays 232, 234, 236 may behave in a similarmanner to the first medium transponders 24, 26, 28 and the first mediumtransponder arrays 104, 200, 202 illustrated in FIG. 36.

FIG. 39 shows details of the N^(TH) wave propagation medium 212illustrated in FIG. 36 according to one embodiment of the N^(TH) wavepropagation medium 212. An N^(TH) medium first transponder 238, anN^(TH) medium second transponder 240, an N^(TH) medium third transponder242, an N^(TH) medium first transponder array 244, an N^(TH) mediumsecond transponder array 246, and an N^(TH) medium third transponderarray 248 are disposed in the N^(TH) wave propagation medium 212 asillustrated in FIG. 39 in a similar manner to the first mediumtransponders 24, 26, 28 and the first medium transponder arrays 104,200, 202 illustrated in FIG. 36. As such, the NTH medium transponders238, 240, 242 are located at the vertices of an equilateral triangle.The N^(TH) medium first transponder array 244 is located between theN^(TH) medium first transponder 238 and the N^(TH) medium secondtransponder 240. The N^(TH) medium second transponder array 246 islocated between the N^(TH) medium first transponder 238 and the N^(TH)medium third transponder 242. The N^(TH) medium third transponder array248 is located between the N^(TH) medium second transponder 240 and theN^(TH) medium third transponder 242. In this regard, the N^(TH) mediumtransponders 238, 240, 242 and the N^(TH) medium transponder arrays 244,246, 248 may behave in a similar manner to the first medium transponders24, 26, 28 and the first medium transponder arrays 104, 200, 202illustrated in FIG. 36.

In general, the polychronous wave propagation system 20 illustrated inFIG. 36 may include the group of wave propagation media 10, 208, 210,212, a corresponding group of initiators for each wave propagationmedium in the group of wave propagation media 10, 208, 210, 212, and acorresponding group of responders for each wave propagation medium inthe group of wave propagation media 10, 208, 210, 212. Eachcorresponding group of initiators is disposed in the corresponding wavepropagation medium in the group of wave propagation media 10, 208, 210,212. Each corresponding group of responders is disposed in thecorresponding wave propagation medium in the group of wave propagationmedia 10, 208, 210, 212. Specifically, the first medium transponders 24,26, 28 and the first medium transponder arrays 104, 200, 202 provide oneof the groups of initiators and one of the groups of responders, both ofwhich are disposed in the first wave propagation medium 10. The secondmedium transponders 214, 216, 218 and the second medium transponderarrays 220, 222, 224 provide one of the groups of initiators and one ofthe groups of responders, both of which are disposed in the second wavepropagation medium 208. The third medium transponders 226, 228, 230 andthe third medium transponder arrays 232, 234, 236 provide one of thegroups of initiators and one of the groups of responders, both of whichare disposed in the third wave propagation medium 210. The N^(TH) mediumtransponders 238, 240, 242 and the N^(TH) medium transponder arrays 244,246, 248 provide one of the groups of initiators and one of the groupsof responders, both of which are disposed in the N^(TH) wave propagationmedium 212.

Any or all of the transponders 24, 26, 28, 214, 216, 218, 226, 228, 230,238, 240, 242 and the transponders in the transponder arrays 104, 200,202, 220, 222, 224, 232, 234, 236, 244, 246, 248 disposed in one of thewave propagation media 10, 208, 210, 212 may be functionally coupled toany or all of the transponders 24, 26, 28, 214, 216, 218, 226, 228, 230,238, 240, 242 and the transponders in the transponder arrays 104, 200,202, 220, 222, 224, 232, 234, 236, 244, 246, 248 disposed in any otherof the wave propagation media 10, 208, 210, 212. As such, anytransponder disposed in one of the wave propagation media 10, 208, 210,212 that detects information from another pair of transponders in thesame wave propagation medium may initiate waves from any transponderdisposed in any other of the wave propagation media 10, 208, 210, 212.By coupling between the wave propagation media 10, 208, 210, 212,flexibility and capacity of the polychronous wave propagation system 20may be increased significantly. Interference between waves may beavoided by dividing waves amongst the wave propagation media 10, 208,210, 212. Storage capacity may be increased by using any or all of thewave propagation media 10, 208, 210, 212 to store information.Resolution of arithmetical computations may be increased by usingmultiple transponder arrays 104, 200, 202, 220, 222, 224, 232, 234, 236,244, 246, 248 across the wave propagation media 10, 208, 210, 212.

In the polychronous wave propagation system 20 illustrated in FIG. 21,the parabolas of intersection 164 are located between the firsttransponder 24 and the second transponder 26. As such, initiation ofwaves from the first transponder 24 and the second transponder 26 occurat times that are less than D/PV. For example, in the polychronous wavepropagation system 20 illustrated in FIG. 28, 0<D/PV, x<D/PV, andy<D/PV. In the polychronous wave propagation system 20 illustrated inFIG. 32, (x+y)<D/PV. However, alternate embodiments of the polychronouswave propagation system 20 may include parabolas of intersection 164outside of the first transponder 24 and the second transponder 26. Sucha polychronous wave propagation system 20 may require a larger firstwave propagation medium 10. Alternatively, the polychronous wavepropagation system 20 illustrated in FIG. 36 may provide a mechanism forcreating parabolas of intersection 164 that are effectively outside thefirst medium transponders 24, 26, 28 by utilizing two or more of thewave propagation media 10, 208, 210, 212. Waves that need to travellonger than D/PV may be used to initiate waves in another medium of thewave propagation media 10, 208, 210, 212. As such, an example ispresented.

Modular arithmetic is a system of arithmetic for integers, in whichnumbers “wrap around” after they reach a certain value, which is calleda modulus. A common example is a 12-hour clock. Every 12 hours the clockwraps around back to zero. As such, if it is presently 7:00, then sevenhours from now will not be 14:00, but will instead be 2:00 since theclock wrapped around to zero at 12:00. Thus, the modulus for a 12-hourclock is 12. In modular arithmetic, if the modulus known, then anynumber may be represented as an integer n and a remainder X. In thisregard, the polychronous wave propagation system 20 illustrated in FIG.36 may be used to perform modular arithmetic and store the result. Themodulus for the polychronous wave propagation system 20 illustrated inFIG. 36 is equal to D/PV. The integer n must be greater than or equal tozero and the remainder X must be less than D/PV. As such, severalequations define the modular arithmetic below.

nD/PV≦x<(n+1)D/PV.  EQ. 21:

X=x−nD/PV.  EQ. 22:

X≡x mod D/PV.  EQ. 23:

EQ. 21 identifies the integer n as the largest integer contained in x.EQ. 22 defines the remainder X as the amount left over after subtractingout the integer n times the modulus D/PV. EQ. 23 defines the remainder Xusing modular arithmetic notation. The steps taken to calculate andstore the integer n and the remainder X are as follows.

Before t=0, all of the transponders 24, 26, 28, 214, 216, 218, 226, 228,230, 238, 240, 242 and all of the transponders in the transponder arrays104, 200, 202, 220, 222, 224, 232, 234, 236, 244, 246, 248 are preventedfrom transmitting waves to allow the polychronous wave propagationsystem 20 to initialize to a stable state.

At t=0, waves are simultaneously initiated from the first mediumtransponders 24, 26, 28.

If t=x is before t=D/PV, then waves are simultaneously initiated fromthe first medium transponders 24, 26, 28. The first medium transponders24, 26, 28 are disabled. When the transponders in the first mediumtransponder arrays 104, 200, 202 simultaneously detect the waves fromthe first medium transponders 24, 26, 28, the transponders that detectedthe waves are indicative of the remainder X and then simultaneouslytransmit waves, thereby storing the remainder X. Since none of thetransponders 24, 26, 28, 214, 216, 218, 226, 228, 230, 238, 240, 242 arestoring information, the integer n is equal to zero.

If t=D/PV=x, then the waves that were initiated at t=0 reach the firstmedium transponders 24, 26, 28, thereby causing waves to besimultaneously initiated from the first medium transponders 24, 26, 28.The first medium transponders 24, 26, 28 continue to initiate waves,thereby storing the integer n. Since the first medium transponders 24,26, 28 are reverberating, the integer n is equal to one. Since none ofthe transponders in the first medium transponder arrays 104, 200, 202are storing information, the remainder X is equal to zero.

If t=D/PV is before t=x, then the waves that were initiated at t=0 reachthe first medium transponders 24, 26, 28, thereby causing waves to besimultaneously initiated from the first medium transponders 24, 26, 28.

If t=x is before t=2D/PV, then waves are simultaneously initiated fromthe first medium transponders 24, 26, 28. When the transponders in thefirst medium transponder arrays 104, 200, 202 simultaneously detect thewaves from the first medium transponders 24, 26, 28, the transpondersthat detected the waves are indicative of the remainder X and thensimultaneously transmit waves, thereby storing the remainder X. Thefirst medium transponders 24, 26, 28 continue to initiate waves, therebystoring the integer n. Since the first medium transponders 24, 26, 28are reverberating, the integer n is equal to one.

If t=2D/PV=x, then the first medium transponders 24, 26, 28 aredisabled. Further, waves are simultaneously initiated from the secondmedium transponders 214, 216, 218. The second medium transponders 214,216, 218 continue to initiate waves, thereby storing the integer n.Since the second medium transponders 214, 216, 218 are reverberating,the integer n is equal to two. Since none of the transponders in thefirst medium transponder arrays 104, 200, 202 are storing information,the remainder X is equal to zero.

If t=2D/PV is before t=x, then the waves that were initiated at t=D/PVreach the first medium transponders 24, 26, 28, thereby causing waves tobe simultaneously initiated from the first medium transponders 24, 26,28.

If t=x is before t=3D/PV, then waves are simultaneously initiated fromthe first medium transponders 24, 26, 28 and the first mediumtransponders 24, 26, 28 are then disabled. Further, waves aresimultaneously initiated from the second medium transponders 214, 216,218. The second medium transponders 214, 216, 218 continue to initiatewaves, thereby storing the integer n. Since the second mediumtransponders 214, 216, 218 are reverberating, the integer n is equal totwo. When the transponders in the first medium transponder arrays 104,200, 202 simultaneously detect the waves from the first mediumtransponders 24, 26, 28, the transponders that detected the waves areindicative of the remainder X and then simultaneously transmit waves,thereby storing the remainder X.

If t=3D/PV=x, then the first medium transponders 24, 26, 28 aredisabled. Further, waves are simultaneously initiated from the thirdmedium transponders 226, 228, 230. The third medium transponders 226,228, 230 continue to initiate waves, thereby storing the integer n.Since the third medium transponders 226, 228, 230 are reverberating, theinteger n is equal to three. Since none of the transponders in the firstmedium transponder arrays 104, 200, 202 are storing information, theremainder X is equal to zero.

In general, the one of the wave propagation media 10, 208, 210, 212 thathas transponders that are reverberating is indicative of the integer n.For example, if the first medium transponders 24, 26, 28 arereverberating, then the integer n is equal to one. If the second mediumtransponders 214, 216, 218 are reverberating, then the integer n isequal to two. If the third medium transponders 226, 228, 230 arereverberating, then the integer n is equal to three. If the N^(TH)medium transponders 238, 240, 242 are reverberating, then the integer nis equal to N. If none of the transponders 24, 26, 28, 214, 216, 218,226, 228, 230, 238, 240, 242 are reverberating, then the integer n isequal to zero. If some of the transponders in the first mediumtransponder arrays 104, 200, 202 are reverberating, then thetransponders in the first medium transponder arrays 104, 200, 202 thatare reverberating are indicative of the remainder X. If none of thetransponders in the first medium transponder arrays 104, 200, 202 arereverberating, then the remainder X is equal to zero.

Alternatively, if the integer n is not needed to be determined and onlythe remainder X is needed to be determined, then the multiple wavepropagation media 10, 208, 210, 212 polychronous wave propagation system20 illustrated in FIG. 36 is not needed. Instead, only the polychronouswave propagation system 20 illustrated in FIG. 33 is needed. The stepstaken to calculate and store the remainder X are as follows.

Before t=0, the first medium transponders 24, 26, 28 and all of thetransponders in the first medium transponder arrays 104, 200, 202 areprevented from transmitting waves to allow the polychronous wavepropagation system 20 to initialize to a stable state.

At t=0, waves are simultaneously initiated from the first mediumtransponders 24, 26, 28.

If t=x is before t=D/PV, then waves are simultaneously initiated fromthe first medium transponders 24, 26, 28. The first medium transponders24, 26, 28 are then disabled. When the transponders in the first mediumtransponder arrays 104, 200, 202 simultaneously detect the waves fromthe first medium transponders 24, 26, 28, the transponders that detectedthe waves are indicative of the remainder X and then simultaneouslytransmit waves, thereby storing the remainder X.

If t=D/PV=x, then the waves that were initiated at t=0 reach the firstmedium transponders 24, 26, 28, thereby causing waves to besimultaneously initiated from the first medium transponders 24, 26, 28.The first medium transponders 24, 26, 28 continue to initiate waves,thereby storing the remainder X as zero.

If t=D/PV is before t=x, then the waves that were initiated at t=0 reachthe first medium transponders 24, 26, 28, thereby causing waves to besimultaneously initiated from the first medium transponders 24, 26, 28.

This process repeats until t=x, such that either the remainder X isstored as zero, in which the first medium transponders 24, 26, 28continue to reverberate waves, or the remainder X is stored as anon-zero value, in which the first medium transponders 24, 26, 28 thatare indicative of the remainder X continue to reverberate and the firstmedium transponders 24, 26, 28 are disabled.

A special case for division using the polychronous wave propagationsystem 20 is division by two. FIG. 40 shows one set of waves transmittedfrom the transponders 24, 26, 28 illustrated in FIG. 33 according to anexemplary embodiment of the polychronous wave propagation system 20. Thefirst set 204 of waves is transmitted at t=0 simultaneously from thetransponders 24, 26, 28. FIG. 40 shows the first set 204 of wavesarriving at transponders in the transponder arrays 104, 200, 202 thatrepresent the parabolas of intersection 164 (not shown) associated withthe middle spatial value ξ_(MV), which occurs when t=D/2, as previouslyshown in FIG. 25. In this regard, the transponders in the transponderarrays 104, 200, 202 that represent the parabolas of intersection 164(not shown) associated with the middle spatial value ξ_(MV) are directlyrepresentative of division by two. Further, each of these transpondersis selected by simultaneously receiving two waves from the first set 204of waves. Due to symmetry in the arrangement of the transponder arrays104, 200, 202, the selected transponders form a reverberating memorycell. In this regard, the results of the division by two may be stored.FIG. 41 shows a set of waves transmitted from the selected transpondersillustrated in FIG. 40 that identify results of the division by twoillustrated in FIG. 40. The waves will re-trigger the selectedtransponders to provide the reverberating memory cell. Division by twomay be useful in manipulations of binary numbers. As such, each mediumin the wave propagation media 10, 208, 210, 212 (FIG. 36) may beassociated with a corresponding binary digit and the results may bestored in a similar manner as shown in FIGS. 40 and 41.

Additional embodiments of the present disclosure are included inAppendix 1 and Appendix 2.

Some of the circuitry previously described may use discrete circuitry,integrated circuitry, programmable circuitry, non-volatile circuitry,volatile circuitry, software executing instructions on computinghardware, firmware executing instructions on computing hardware, thelike, or any combination thereof. The computing hardware may includemainframes, micro-processors, micro-controllers, DSPs, the like, or anycombination thereof.

None of the embodiments of the present disclosure are intended to limitthe scope of any other embodiment of the present disclosure. Any or allof any embodiment of the present disclosure may be combined with any orall of any other embodiment of the present disclosure to create newembodiments of the present disclosure.

Those skilled in the art will recognize improvements and modificationsto the preferred embodiments of the present invention. All suchimprovements and modifications are considered within the scope of theconcepts disclosed herein and the claims that follow.

What is claimed is:
 1. A polychronous wave propagation system comprisinga first plurality of initiators disposed in a first wave propagationmedium and a first plurality of responders disposed in the first wavepropagation medium, such that: the first plurality of initiators areadapted to transmit at least two propagated waves and the firstplurality of responders are adapted to receive the propagated waves;relative timing is associated with interference patterns of energybetween the propagated waves; and operational behavior of thepolychronous wave propagation system is based on the relative timing anddistances between the first plurality of initiators and the secondplurality of initiators.
 2. The polychronous wave propagation system ofclaim 1 wherein: the propagated waves have encoded information; at leasta first parabola of intersection formed in the first wave propagationmedium is based on timing of transmission of the encoded informationfrom each of a first pair of the first plurality of initiators; the atleast the first parabola of intersection is representative of anarithmetical relationship between the transmission of the encodedinformation from each of the first pair of the first plurality ofinitiators; a corresponding one of the first plurality of responders islocated on each of the at least the first parabola of intersection; eachcorresponding one of the first plurality of responders is adapted todetect information based on timing between reception of the encodedinformation from each of the first pair of the first plurality ofinitiators; and operational behavior of the polychronous wavepropagation system is based on the detected information.
 3. Thepolychronous wave propagation system of claim 2 wherein the detectedinformation associated with the one of the first plurality of respondersis further based on simultaneous reception of the encoded informationfrom the first pair of the first plurality of initiators.
 4. Thepolychronous wave propagation system of claim 2 wherein the operationalbehavior includes arithmetical computations.
 5. The polychronous wavepropagation system of claim 4 wherein the arithmetical computationscomprise additions.
 6. The polychronous wave propagation system of claim5 wherein: a first transmission of encoded information is sent from thefirst pair of the first plurality of initiators; timing of a secondtransmission of encoded information from the first pair of the firstplurality of initiators is based on a first magnitude of a firstvariable and a second magnitude of a second variable; and one of the atleast the first parabola of intersection is based on a sum between thefirst magnitude and the second magnitude.
 7. The polychronous wavepropagation system of claim 4 wherein the arithmetical computationscomprise subtractions.
 8. The polychronous wave propagation system ofclaim 7 wherein: timing of a first transmission of encoded informationfrom the first pair of the first plurality of initiators is based on afirst magnitude of a first variable; timing of a second transmission ofencoded information from the first pair of the first plurality ofinitiators is based on a second magnitude of a second variable; and oneof the at least the first parabola of intersection is based on adifference between the first magnitude and the second magnitude.
 9. Thepolychronous wave propagation system of claim 4 wherein the arithmeticalcomputations comprise multiplications.
 10. The polychronous wavepropagation system of claim 9 wherein: timing of a first transmission ofencoded information from the first pair of the first plurality ofinitiators is based on a first magnitude of a first variable; timing ofa second transmission of encoded information from the first pair of thefirst plurality of initiators is based on a second magnitude of a secondvariable; and one of the at least the first parabola of intersection isbased on a product between the first magnitude and the second magnitude.11. The polychronous wave propagation system of claim 4 wherein thearithmetical computations comprise divisions.
 12. The polychronous wavepropagation system of claim 11 wherein: timing of a first transmissionof encoded information from the first pair of the first plurality ofinitiators is based on a first magnitude of a first variable; timing ofa second transmission of encoded information from the first pair of thefirst plurality of initiators is based on a second magnitude of a secondvariable; and one of the at least the first parabola of intersection isbased on a quotient between the first magnitude and the secondmagnitude.
 13. The polychronous wave propagation system of claim 12wherein the quotient is based on the first magnitude divided by thesecond magnitude.
 14. The polychronous wave propagation system of claim1 further comprising a plurality of wave propagation media, acorresponding plurality of initiators for each of the plurality of wavepropagation media, and a corresponding plurality of responders for eachof the plurality of wave propagation media, such that: eachcorresponding plurality of initiators is disposed in a corresponding oneof the plurality of wave propagation media; each corresponding pluralityof responders is disposed in a corresponding one of the plurality ofwave propagation media; the plurality of wave propagation mediacomprises the first wave propagation medium; the corresponding pluralityof initiators for the first wave propagation medium comprises the firstplurality of initiators; and the corresponding plurality of respondersfor the first wave propagation medium comprises the first plurality ofresponders.
 15. The polychronous wave propagation system of claim 1wherein the operational behavior is representative of at least onereverberating memory cell.
 16. The polychronous wave propagation systemof claim 15 wherein the operational behavior includes modulararithmetic.
 17. The polychronous wave propagation system of claim 1further comprising a control system adapted to receive responderinformation and provide initiator control information, such that timingof transmission of the encoded information from each of at least one ofthe first plurality of initiators is based on the initiator controlinformation, and each of at least one of the first plurality ofresponders is adapted to provide the responder information based oncorresponding detected information from each of the at least one of thefirst plurality of responders.
 18. The polychronous wave propagationsystem of claim 1 wherein the first wave propagation medium comprises asolid material.
 19. The polychronous wave propagation system of claim 1wherein a longest dimension of the first wave propagation medium is lessthan about one meter.
 20. The polychronous wave propagation system ofclaim 1 wherein the first wave propagation medium forms anano-structure.
 21. The polychronous wave propagation system of claim 1wherein the first plurality of responders and the first plurality ofinitiators are disposed in fixed positions in the first wave propagationmedium.
 22. The polychronous wave propagation system of claim 1 whereinthe polychronous wave propagation system is adapted to providepolychronous activity that is analogous to polychronous activity in abrain.
 23. The polychronous wave propagation system of claim 1 furthercomprising a second plurality of initiators disposed in the first wavepropagation medium and a second plurality of responders disposed in thefirst wave propagation medium, such that: the second plurality ofinitiators are adapted to simultaneously transmit a set of at leastthree propagated waves and the second plurality of responders areadapted to receive the set of at least three propagated waves; and uponreceipt of the set of at least three propagated waves by the secondplurality of responders, the second plurality of initiators are adaptedto simultaneously re-transmit the set of at least three propagated wavesto provide a reverberating memory cell.
 24. The polychronous wavepropagation system of claim 23 wherein the second plurality ofresponders is identical to the first plurality of responders.
 25. Thepolychronous wave propagation system of claim 23 wherein the operationalbehavior includes an arithmetical computation and the reverberatingmemory cell stores results of the arithmetical computation.
 26. A methodcomprising: forming a polychronous wave propagation system by providinga first plurality of initiators disposed in a first wave propagationmedium and a first plurality of responders disposed in the first wavepropagation medium; transmitting at least two propagated waves; andreceiving the propagated waves, such that: relative timing is associatedwith interference patterns of energy between the propagated waves; andoperational behavior of the polychronous wave propagation system isbased on the relative timing and distances between the first pluralityof initiators and the second plurality of initiators.